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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Population growth 41. When records were first kept (t = 0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P(t)=30(1+t), where t is measured in years. a. What is the population after 20 years? b. Find the population P(t) at any time t 0.Population growth 42. The population of a community of foxes is observed to fluctuate on a 10-year cycle due to variations in the availability of prey. When population measurements began (t = 0), the population was 35 foxes. The growth rate in units of foxes/year was observed to be P(t)=5+10sint5. a. What is the population 15 years later? 35 years later? b. Find the population P(t) at any time t 0.Population growth 43. A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N(t) = 100e0.25t. Assume t is measured in days. a. What is the population after 20 days? After 40 days? b. Find the population N(t) at any time t 0.Cancer treatment A cancerous tumor in a mouse is treated with a chemotherapy drug After treatment, the rate of change in the size of the tumor (in cm3/day) is given by the function r(t) = 0.0025e0.25t 0.1485e 0.15t, where t is measured in days. a. Find the value of t0 for which r(t0) = 0. b. Plot the function r(t), for 0 t 15, and describe what happens to the tumor over the 15-day period after treatment. c. Evaluate 0t0r(t)dt and interpret the physical meaning of this integral.Oil production An oil refinery produces oil at a variable rate given by Q(t)={800if0t30260060tif30t40200ift40, where t is measured in days and Q is measured in barrels. a. How many barrels are produced in the first 35 days? b. How many barrels are produced in the first 50 days? c. Without using integration, determine the number of barrels produced over the interval [60, 80].Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions r1(t)=0.25t2+37.46t+722.47(April)andr2(t)=0.90t269.06t+2053.12(June), where the discharge is measured in millions of cubic feet per day and t = 1 corresponds to the first day of the month (see figure). a. Determine the total amount of water that flows through Spokane in April (30 days). b. Determine the total amount of water that flows through Spokane in June (30 days). c. The Spokane River flows out of Lake Coeur dAlene, which contains approximately 0.67 mi3 of water. Determine the percentage of Lake Coeur dAlenes volume that flows through Spokane in April and June.Depletion of natural resources Suppose that r(t) = r0 ekt, with k 0, is the rate at which a nation extracts oil, where r0 = 107 barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2 109 barrels. a. Find Q(t), the total amount of oil extracted by the nation after t years. b. Evaluate limtQ(t) and explain the meaning of this limit. c. Find the minimum decay constant k for which the total oil reserves will last forever. d. Suppose r0 = 2 107 barrels/yr and the decay constant k is the minimum value found in part (c). How long will the total oil reserves last?Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t = 0) at a rate (in L/min) given by Q(t)=3t, where t is measured in minutes. a. How much water flows into the cistern in 1 hour? b. Find and graph the function that gives the amount of water in the tank at any time t 0. c. When will the tank be full?Filling a reservoir A reservoir with a capacity of 2500 m3 is filled with a single inflow pipe. The reservoir is empty when the inflow pipe is opened at t = 0. Letting Q(t) be the amount of water in the reservoir at time t, the flow rate of water into the reservoir (in m3/hr) oscillates on a 24-hr cycle (see figure) and is given by Q(t)=20(1+cost12). a. How much water flows into the reservoir in the first 2 hr? b. Find the function that gives the amount of water in the reservoir over the interval [0. t], where t 0. c. When is the reservoir full?Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V(t) = 70(1 + sin 2t), where V(t) is the amount of blood (in millilitres) pumped over the interval [0, t], V(0) = 0, and t is measured in seconds. a. Verify that the amount of blood pumped over a one-second interval s 70 mL. b. Find the function that gives the total blood pumped between t = 0 and a future time t 0. c. What is the cardiac output over a period of 1 min? (Use calculus: then check your answer with algebra.)Air flow in the lungs A simple model (with different parameters for different people) for the flow of air in and out of the lungs is V(t)=2sint2, where V(t) (measured in liters) is the volume of air in the lungs at time t 0, t is measured in seconds, and t = 0 corresponds to a time at which the lungs are full and exhalation begins. Only a fraction of the air in the lungs is exchanged with each breath. The amount that is exchanged is called the tidal volume. a. Find the volume function V assuming V(0) = 6 L. b. What is the breathing rate in breaths/min? c. What i3 the tidal volume and what is the total capacity of the lungs?Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N(t)=r+Asin2tP where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t = 0. a. Suppose P = 10, A = 20, and r = 0. If the initial population is N(0) = 10, does the population ever become extinct? Explain. b. Suppose P = 10, A = 20, and r = 0. If the initial population is N(0) = 100, does the population ever become extinct? Explain. c. Suppose P = 10, A = 50, and r = 5. If the initial population is N(0) = 10, does the population ever become extinct? Explain. d. Suppose P = 10, A = 50, and r = 5. Find the initial population N(0) needed to ensure that the population never becomes extinct.Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal = 4184 J. One hour of walking consumes roughly 106 J, or 250 Cal. On the other hand, power is the rate at which energy is used and is measured in watts (W; 1 W = 1 J/s). Other useful units of power are kilowatts (1 kW = 103 W) and megawatts (1 MW = 106 W). If energy is used at a rate of 1 kW for 1 hr, the total amount of energy used is 1 kilowatt-hour (kWh), which is 3.6 106 J. Suppose the power function of a large city over a 24-hr period is given by P(t)=E(t)=300200sint12, where P is measured in megawatts and t = 0 corresponds to 6:00 P.M. (see figure). a. How much energy is consumed by this city in a typical 24-hr period? Express the answer in megawatt-hours and in joules. b. Burning 1 kg of coal produces about 450 kWh of energy. How many kg of coal are required to meet the energy needs of the city for 1 day? For 1 year? c. Fission of 1 g of uranium-235 (U-235) produces about 16,000 kWh of energy. How many grams of uranium are needed to meet the energy needs of the city for 1 day? For 1 year? d. A typical wind turbine can generate electrical power at a rate of about 200 kW. Approximately how many wind turbines are needed to meet the average energy needs of the city?Carbon uptake An important process in the study of global warming and greenhouse gases is the net ecosystem exchange, which is the rate at which carbon leaves an ecosystem and enters the atmosphere in a particular geographic region. Let N(t) equal the net ecosystem exchange on an average July day in a high-altitude coniferous forest, where N(t) is measured in grams of carbon per square meter per hour and f is the number of hours past midnight so that 0 t 24 (see figure). Negative values of N correspond to times when the amount of carbon in the atmosphere decreases, and positive values of N occur when the amount of carbon in the atmosphere increases. a.Trees and other plants help reduce carbon emissions in the atmosphere by using photosynthesis to absorb carbon dioxide and release oxygen Give a possible explanation why N is negative on the interval 5 t 17. b.The cumulative net carbon uptake, 024N(t)dt, is the net change in the amount of carbon in the atmosphere over a 24-hour period in the coniferous forest Use a midpoint Riemann sum with n = 12 subintervals to estimate the cumulative net carbon uptake in the coniferous forest and interpret the resultMarginal cost Consider the following marginal cost functions. a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units. b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units. 45. C(x) = 2000 0.5xMarginal cost Consider the following marginal cost functions. a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units. b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units. 46. C(x) = 200 0.05xMarginal cost Consider the following marginal cost functions. a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units. b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units. 47. C(x) = 300 + 10x 0.01x258EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The distance traveled by an object moving along a line is the same as the displacement of the object. b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal. c. Consider a tank that is filled and drained at a flow rate of V(t) = 1 t2/100 (gal/min), for t 0, where t is measured in minutes. It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty. d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from A units to 2A units is greater than the cost of increasing production from 2A units to 3A units.Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of _____. 52. v(t) = 2t + 6, for 0 t 8Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of _____. 53. v(t) = 1 t2/16, for 0 t 4Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of _____. 54. v(t) = 2 sin t, for 0 tEquivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of _____. 55. v(t) = t(25 t2)1/2, for 0 t 5Where do they meet? Kelly started at noon (t = 0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15/(t + 1)2 (decreasing because of fatigue). Sandy started at noon (t = 0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20/(t + 1)2 (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours. a. Make a graph of Kellys distance from Niwot as a function of time. b. Make a graph of Sandys distance from Berthoud as a function of time. c. When do they meet? How far has each person traveled when they meet? d. More generally, if the riders speeds are v(t) = A/(t + 1)2 and u(t) = B/(t + 1)2 and the distance between the towns is D, what conditions on A, B, and D must be met to ensure that the riders will pass each other? e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).65ETwo runners At noon (t = 0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4/(t + 1), for t 0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2/(t + 1), for t 0. Assume t is measured in hours. a. Find the position functions for Alicia and Boris, where s = 0 corresponds to Alicias starting point. b. When, if ever, does Alicia overtake Boris?Snowplow problem With snow on the ground and falling at a constant rate, a snowplow began plowing down a long straight road at noon. The plow traveled twice as far in the first hour as it did in the second hour. At what time did the snow start falling? Assume the plowing rate is inversely proportional to the depth of the snow.Variable gravity At Earths surface, the acceleration due to gravity is approximately g = 9.8 m/s2 (with local variations). However, the acceleration decreases with distance from the surface according to Newtons law of gravitation. At a distance of y meters from Earths surface, the acceleration is given by a(y)=g(1+y/R)2, where R = 6.4 106 m is the radius of Earth. a. Suppose a projectile is launched upward with an initial velocity of v0 m/s. Let v(t) be its velocity and y(t) its height (in meters) above the surface t seconds after the launch. Neglecting forces such as air resistance, explain why dvdt=a(y) and dydt=v(t). b. Use the Chain Rule to show that dvdt=12ddy(v2). c. Show that the equation of motion for the projectile is 12ddy(v2)=a(y), where a(y) is given previously. d. Integrate both sides of the equation in part (c) with respect to y using the fact that when y = 0, v = v0. Show that 12(v2v02)=gR(11+y/R1). e. When the projectile reaches its maximum height, v = 0 Use this fact to determine that the maximum height is ymax=Rv022gRv02. f. Graph ymax as a function of v0. What is the maximum height when v0 = 500 m/s, 1500 m/s, and 5 km/s? g. Show that the value of v0 needed to put the projectile into orbit (called the escape velocity) is 2gR.Another look at the Fundamental Theorem 69. Suppose that f and g have continuous derivatives on an interval [a, b]. Prove that if f(a) = g(a) and f(b) = g(b), then abf(x)dx=abg(x)dx.Another look at the Fundamental Theorem 70. Use Exercise 69 to prove that if two runners start and finish at the same time and place, then regardless of the velocities at which they run, their displacements are equal. 69. Suppose that f and g have continuous derivatives on an interval [a, b]. Prove that if f(a) = g(a) and f(b) = g(b), then abf(x)dx=abg(x)dx.Another look at the Fundamental Theorem 71. Use Exercise 69 to prove that if two trails start at the same place and finish at the same place, then regardless of the ups and downs of the trails, they have the same net change in elevation. 69. Suppose that f and g have continuous derivatives on an interval [a, b]. Prove that if f(a) = g(a) and f(b) = g(b), then abf(x)dx=abg(x)dx.Another look at the Fundamental Theorem 72. Without evaluating integrals, prove that 02ddx(12sinx2)dx=02ddx(x10(2x)3)dx.In the area formula for a region between two curves, verify that if the lower curve is g(x) = 0, the formula becomes the usual formula for the area of the region bounded by y = f(x) and the x-axis.Interpret the area formula when it is written in the form A=abf(x)dxabg(x)dx, where f(x)g(x)0 on [a, b].The region R is bounded by the curve y=x the line y = x 2, and the x-axis. Express the area of R in terms of (a) integral(s) with respect to x and (b) integral(s) with respect to y.An alternative way to determine the area of the region in Example 3 (Figure 6.18) is to compute 02(x+6x2)dx. Why? Example 3 Integrating with Respect to y Find the area of the region R bounded by the graphs of y=x3, y = x + 6, and the x-axis.Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on [a, c] (see figure). Assume the curves intersect at x = bSet up an integral that equals the area of the region (see figure) in the following two ways. Do not evaluate the integrals. a.Using integration with respect to x. b.Using integration with respect to y.Make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to x.Make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to y.Find the area of the region (see figure) in two ways. a.Using integration with respect to x. b.Using geometry.Find the area of the region (see figure) in two ways. a.By integrating with respect to y. b.Using geometryExpress the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals. 5. Find the area of the region (see figure) in two ways. a.Using integration with respect to x. b.Using geometry.Express the area of the shaded region in Exercise 6 as the sum of two integrals with respect to x. Do not evaluate the integrals. 6.Find the area of the region (see figure) in two ways. a.By integrating with respect to y. b.Using geometry.Finding area Determine the area of the shaded region in the following figures. 5.Finding area Determine the area of the shaded region in the following figures. 10.Finding area Determine the area of the shaded region in the following figures. 11.Finding area Determine the area of the shaded region in the following figures. 12.Finding area Determine the area of the shaded region in the following figures. 7.Finding area Determine the area of the shaded region in the following figures. 14.Finding area Determine the area of the shaded region in the following figures. 15.Finding area Determine the area of the shaded region in the following figures. 6.Finding area Determine the area of the shaded region in the following figures. 17.Finding area Determine the area of the shaded region in the following figures. 8. (Hint: Find the intersection point by inspection.)Finding area Determine the area of the shaded region in the following figures. 19.Finding area Determine the area of the shaded region in the following figures. 20.Finding area Determine the area of the shaded region in the following figures. 21.Finding area Determine the area of the shaded region in the following figures. 22.Finding area Determine the area of the shaded region in the following figures. 23.Finding area Determine the area of the shaded region in the following figures. 24.Finding area Determine the area of the shaded region in the following figures. 25.Finding area Determine the area of the shaded region in the following figures. 26.Finding area Determine the area of the shaded region in the following figures. 27.Finding area Determine the area of the shaded region in the following figures. 28.Finding area Determine the area of the shaded region in the following figures. 29.Finding area Determine the area of the shaded region in the following figures. 30.Two approaches Express the area of the following shaded regions in terms of (a) one or more integrals with respect to x and (b) one or more integrals with respect to y. You do not need to evaluate the integrals. 30.Two approaches Express the area of the following shaded regions in terms of (a) one or more integrals with respect to x and (b) one or more integrals with respect to y. You do not need to evaluate the integrals. 28.Area between velocity curves Two runners, starting at the same location, run along a straight road for 1 hour. The velocity of ore runner is v1(t) 7t and the velocity of the other rurner is v2(t)10t. Assume t is measured in hou'S and the veocities v1(t) and v2(t) are measured in km/hr. Determine the area between the curves y = v1 (l) and y = v2(l), for 0 l 1. Interpret the physical meaning of this areaCalculus and geometry For the given regions R1 and R2, complete the following steps. a.Find the area of region R1. b.Find the area of region R2 using geometry and the answer to part (a). 34.R1 is the region in the first quadrant bounded by the y-axis and the curves y = 2x2 and y = 3 = x; R2 is the region in the first quadrant bounded by the x-axis and the curves y = 2x2 and y = 3 x (see figure).Calculus and geometry For the given regions R1 and R2, complete the following steps. a.Find the area of region R1. b.Find the area of region R2 using geometry and the answer to part (a). 35.R1 is the region in the first quadrant bounded by the line x = 1 and the curve y = 6x(2 x2)2; R2 is the region in the first quadrant bounded the curve y = 6x(2 x2)2 and the line y = 6x.Calculus and geometry For the given regions R1 and R2, complete the following steps. a.Find the area of region R1. b.Find the area of region R2 using geometry and the answer to part (a). 36.R1 is the region in the first quadrant bounded by the coordinate axes and the curve y = cos1x; R2 is the region bounded by the lines y=2 and x = 1, and the curve y = cos1x.Regions between curves Find the area of the region described in the following exercises. 37.The region bounded by y=4x+4, y=6x+6,and x = 4Regions between curves Find the area of the region described in the following exercises. 38.The region bounded by y = cos x and y = sin x between x=4 and x=54Regions between curves Find the area of the region described in the following exercises. 39.The region bounded by y = ex, and y = e 2x and x=ln4Regions between curves Find the area of the region described in the following exercises. 40.The region bounded by y = 6x and y = 3x2 6xRegions between curves Find the area of the region described in the following exercises. 41.The region bounded by y=21+x2 and y=1Regions between curves Find the area of the region described in the following exercises. 42.The region bounded by y=24x and y = 3x2Regions between curves Find the area of the region described in the following exercises. 43.The region bounded by y = x, y=1x, y = 0and x = 2Regions between curves Find the area of the region described in the following exercises. 44.The region in the first quadrant on the interval [0, 2] bounded by y=4xx2 and y=4x4Regions between curves Find the area of the region described in the following exercises. 45.The region bounded by y = 2 |x| and y = x2Regions between curves Find the area of the region described in the following exercises. 46.The region bounded by y = x3 and y = 9xRegions between curves Find the area of the region described in the following exercises. 47.The region bounded by y = |x 3| and y=x2Regions between curves Find the area of the region described in the following exercises. 48.The region bounded by y = 3x x3 and y = xAny method Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves and the region in question. 33. The region in the first quadrant bounded by y = x2/3 and y = 4Regions between curves Find the area of the region described in the following exercises. 50.The region in the first quadrant bounded by y = 2 and y = 2 sin x on the interval [0, /2]Regions between curves Find the area of the region described in the following exercises. 51.The region bounded by y = ex, y=2ex+1 and x = 052ERegions between curves Find the area of the region described in the following exercises. 53.The region between the line y = x and the curve y=2x1x2 in the first quadrantRegions between curves Find the area of the region described in the following exercises. 54.The region bounded by x=y24 andy=x3Regions between curves Find the area of the region described in the following exercises. 55.The region bounded by y=x, y = 2x 15, and y = 0Regions between curves Find the area of the region described in the following exercises. 56.The region bounded by y = 2 andy=11x2Regions between curves Find the area of the region described in the following exercises. 57.The region bounded by y=x22x+1 andy=5x9Regions between curves Find the area of the region described in the following exercises. 58.The region bounded by x=y(y1) andx=y(y1)Regions between curves Find the area of the region described in the following exercises. 59.The region bounded by x=y(y1) andy=x360ERegions between curves Find the area of the region described in the following exercises. 61.The region in the first quadrant bounded by y=521x and y = xRegions between curves Find the area of the region described in the following exercises. 62.The region in the quadrant bounded by y=x1, y = 4x, and y=x4Complicated regions Find the area of the regions shown in the following figures. 53.Complicated regions Find the area of the regions shown in the following figures. 52.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The area of the region bounded by y = x and x = y2 can be found only by integrating with respect to x. b. The area of the region between y = sin x and y = cos x on the interval [0, /2] is 0/2(cosxsinx)dx. c. 01(xx2)dx=01(yy)dy.Differences of even functions Assume f and g are even, integrable functions on [a, a], where a 1. Suppose f(x) g(x) 0 on [a, a] and the area bounded by the graphs of f and g on [a, a] is 10. What is the value of 0ax(f(x2)g(x2))dx? 73. Roots and powers Consider the functions f(x) = xn and g(x) = x1/n, where n 2 is a positive integer. a. Graph f and g for n = 2, 3, and 4, for x 0. b. Give a geometric interpretation of the area function An(x)=0x(f(s)g(s))ds, for n = 2, 3, 4, and x 0. c. Find the positive root of An(x) = 0 in terms of n. Does the root increase or decrease with n?Area of a curve defined implicitly Determine the area of the shaded region bounded by the curve x2 = y4(1 y3) (see figure).68E69E70E71E72EBisecting regions For each region R, find the horizontal line y = k that divides R into two subregions of equal area. 61. R is the region bounded by y = 1 |x 1| and the x-axis.Geometric probability Suppose a dartboard occupies the square {(x, y):0 |x| 1, 0 |y| 1}. A dart is thrown randomly at the board many times (meaning it is equally likely to land at any point in the square). What fraction of the dart throws land closer to the edge of the board than the center? Equivalently, what is the probability that the dart lands closer to the edge of the board than the center? Proceed as follows. a. Argue that by symmetry, it is necessary to consider only one quarter of the board, say the region R: {(x, y): |x| y 1}. b. Find the curve C in this region that is equidistant from the center of the board and the top edge of the board (see figure). c. The probability that the dart lands closer to the edge of the board than the center is the ratio of the area of the region R1 above C to the area of the entire region R. Compute this probability.Lorenz curves and the Gini index A Lorenz curve is given by y = L(x), where 0 x 1 represents the lowest fraction of the population of a society in terms of wealth and 0 y 1 represents the fraction of the total wealth that is owned by that fraction of the society. For example, the Lorenz curve in the figure shows that L(0.5) = 0.2, which means that the lowest 0.5 (50%) of the society owns 0.2 (20%) of the wealth. (See the Guided Project Distribution of Wealth for more on Lorenz curves.) a. A Lorenz curve y = L(x) is accompanied by the line y = x, called the line of perfect equality. Explain why this line is given this name. b. Explain why a Lorenz curve satisfies the conditions L(0) = 0, L(1) = 1, L(x) x, and L(x) 0 on [0, 1]. c. Graph the Lorenz curves L(x) = xp corresponding to p = 1.1, 1.5, 2, 3, 4. Which value of p corresponds to the most equitable distribution of wealth (closest to the line of perfect equality)? Which value of p corresponds to the least equitable distribution of wealth? Explain. d. The information in the Lorenz curve is often summarized in a single measure called the Gini index, which is defined as follows. Let A be the area of the region between y = x and y = L(x) (see figure) and let B be the area of the region between y = L(x) and the x-axis. Then the Gini index is G=AA+B. Show that G=2A=1201L(x)dx. e. Compute the Gini index for the cases L(x) = xp and p = 1.1, 1.5, 2, 3, 4. f. What is the smallest interval [a, b] on which values of the Gini index lie for L(x) = xp with p 1? Which endpoints of [a, b] correspond to the least and most equitable distribution of wealth? g. Consider the Lorenz curve described by L(x) = 5x2/6 + x/6. Show that it satisfies the conditions L(0) = 0, L(1) = 1, and L(x) 0 on [0, 1]. Find the Gini index for this function.Equal area properties for parabolas Consider the parabola y = x2. Let P, Q, and R be points on the parabola with R between P and Q on the curve. Let P, Q, and R be the lines tangent to the parabola at P, Q, and R, respectively (see figure). Let P be the intersection point of Q and R, let Q be the intersection point of P and R, and let R be the intersection point of P and Q. Prove that Area PQR = 2 Area PQR in the following cases. (In fact, the property holds for any three points on any parabola.) (Source: Mathematics Magazine 81, 2, Apr 2008) a. P(a, a2), Q(a, a2), and R(0, 0), where a is a positive real number b. P(a, a2), Q(b, b2), and R(0, 0), where a and b are positive real numbers c. P(a, a2), Q(b, b2), and R is any point between P and Q on the curve77EShifting sines Consider the functions f(x) = a sin 2x and g(x) = (sin x)/a, where a 0 is a real number. a. Graph the two functions on the interval [0, /2], for a=12, 1, and 2. b. Show that the curves have an intersection point x (other than x = 0) on [0, /2] that satisfies cos x 1/(2a2), provided a1/2. c. Find the area of the region between the two curves on [0, x] when a = 1. d. Show that as a1/2+, the area of the region between the two curves on [0, x] approaches zero.Why is the volume as given by the general slicing method, equal to the average value of the area function A on [a, b] multiplied by b a?In Example 2 what is the cross-sectional area function A(x) if cross sections perpendicular to the base are squares rather than semicircles? Example 2 Volume of a Parabolic Hemisphere A solid has a base that is bounded by the curves y = x2 and y = 2 x2 in the xy-plane. Cross sections through the solid perpendicular to the base and parallel to the y-axis are semicircular disks. Find the volume of the solid.What solid results when the region R is revolved about the x-axis if (a) R is a square with vertices (0, 0), (0, 2), (2, 0), and (2, 2), and (b) R is a triangle with vertices (0, 0), (0, 2), and (2, 0)?
Show that when g(x) = 0 in the washer method, the result is the disk method.Suppose the region in Example 4 is revolved about the line y = 1 instead of the x-axis. (a) What is the inner radius of a typical washer? (b) What is the outer radius of a typical washer? Example 4 Volume by the Washer Method The region R is bounded by the graphs of f(x)x and g(x) x2 between x 0 and x 1. What is the volume of the solid that results when R s revolved about the x-axis?The region in the first quadrant bounded by y = x and y = x3 is revolved about the y-axis Give the integral for the volume of the solid that is generated.Suppose a cut is made through a solid object perpendicular to the x-axis at a particular point x. Explain the meaning of A(x).A solid has a circular base and cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?Consider a solid whose base is the region in the first quadrant bounded by the curve y=3x and the line x = 2, and whose cross sections through the solid perpendicular to the x-axis are squares. a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2]. b. Write an integral for the volume of the solid.Why is the disk method a special case of the general slicing method?Let R be the region bounded by the curve y=cosx and the x-axis on [0, /2]. A solid of revolution is obtained by revolving R about the x-axis (see figures). a. Find an expression for the radius of a cross section of the solid of revolution at a point x in [0, /2]. b. Find an expression for the area A(x) of a cross section of the solid at a point x in [0. /2]. c. Write an integral for the volume of the solid.Let R be the region bounded by the curve y = cos1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures). a. Find an expression for the radius of a cross section of the solid at a point y in [0, /2]. b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0, /2] c. Write an integral for the volume of the solid.Use the region R that is bounded by the graphs of y=1+x, x = 4, and y = 1 to complete the exercises. 7.Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers a. What is the outer radius of a cross section of the solid at a point x in [0, 4]? b. What is the inner radius of a cross section of the solid at a point x in [0, 4]? c. What is the area A(x) of a cross section of the solid at a point x in [0, 4]? d. Write an integral for the volume of the solid.Use the region R that is bounded by the graphs of y=1+x, x = 4, and y = 1 to complete the exercises. 8.Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers a.What is the outer radius of a cross section of the solid at a point y in [1, 3]? b.What is the inner radius of a cross section of the solid at a point y in [1, 3]? c.What is the area A(y) of a cross section of the solid at a point y in [1, 3]? d.Write an integral for the volume of the solid.Use the region R that is bounded by the graphs of y=1+x, x = 4, and y = 1 to complete the exercises. 9.Region R is revolved about the line y = 1 to form a solid of revolution. a.What is the radius of a cross section of the solid at a point x in [0, 4]? b.What is the area A(x) of a cross section of the solid at a point x in [0, 4]? c.Write an integral for the volume of the solid.Use the region R that is bounded by the graphs of y=1+x, x = 4, and y = 1 to complete the exercises. 10.Region R is revolved about the line x = 4 to form a solid of revolution. a.What is the radius of a cross section of the solid at a point y in [1, 3]? b.What is the area A(y) of a cross section of the solid at a point y in [1, 3]? c.Write an integral for the volume of the solid.General slicing method Use the general slicing method to find the volume of the following solids. 8. The solid whose base is the region bounded by the semicircle y=1x2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squaresGeneral slicing method Use the general slicing method to find the volume of the following solids. 7. The solid whose base is the region bounded by the curves y = x2 and y = 2 x2, and whose cross sections through the solid perpendicular to the x-axis are squaresGeneral slicing method Use the general slicing method to find the volume of the following solids. 9. The solid whose base is the region bounded by the curve y=cosx and the x-axis on [/2, /2], and whose cross sections through the solid perpendicular to the x-axis are isosceles right triangles with a horizontal leg in the xy-plane and a vertical leg above the x-axisGeneral slicing method Use the general slicing method to find the volume of the following solids. 11. The solid with a semicircular base of radius 5 whose cross sections perpendicular to the base and parallel to the diameter are squaresGeneral slicing method Use the general slicing method to find the volume of the following solids. 13. The solid whose base is the triangle with vertices (0, 0), (2, 0), and (0, 2), and whose cross sections perpendicular to the base and parallel to the y-axis are semicirclesGeneral slicing method Use the general slicing method to find the volume of the following solids. 16.The solid whose base is the region bounded by y = x2 and the line y = 1, and whose cross sections perpendicular to the base and parallel to the y-axis are squaresDisk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis. 17. y = 2x, y = 0, x = 3 (Verify that your answer agrees with the volume formula for a cone.)Disk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis. 18. y = 2 2x, y = 0, x = 0 (Verify that your answer agrees with the volume formula for a cone.)Disk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis. 19. y = ex, y = 0, x = 0, x = ln 4Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 20.y=11x24,y = 0, x = 0, and x=12; about the x axisSolids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 21.x=11+y2,x = 0, y = 1, and y = 1; about the y-axisDisks/washers about the y-axis Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis. 36. y = 0, y = ln x, y = 2, x = 0Washer method Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis. 27. y = x, y=2xWasher method Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis. 28. y = x, y=x4Washer method Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis. 29. y = ex/2, y = ex/2, x = ln 2, x = ln 3Washer method Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis. 30. y = x, y = x + 2, x = 0, x = 4Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 27.y=x3,y = 0, and x = 1; about the y-axisDisks/washers about the y-axis Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis. 35. y = x, y = 2x, y = 6Disk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis. 20. y = cos x on [0, /2], y = 0, x = 0 (Recall that cos2x=12(1+cos2x).)Disk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis. 22. y=25x2, y = 0 (Verify that your answer agrees with the volume formula for a sphere.)Disk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis. 21. y = sin x on [0, ], y = 0 (Recall that sin2x=12(1cos2x).)Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 32.y = sec 1 x, = 0. x = 0, y = 0, and y=4; about the y-axisSolids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 33.y=sin1 x, x = 0,y=4; about the y-axisWasher method Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis. 32. y=sinx, y = 1, x = 0Washer method Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis. 33. y = sin x, y=sinx, for 0 x /2Washer method Let R be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when R is revolved about the x-axis. 34. y = |x|, y = 2 x2Disks/washers about the y-axis Let R be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when R is revolved about the y-axis. 38. y=x, y = 0, x = 4Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 38.y=4x2, x = 2, and y = 4; and the y-axisSolids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 39.y=1/x, y = 0, x = 2, and x = 6; about the x-axisSolids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 40.y x2, y 2 x, and y 0; about the y-axisSolids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 41.y = ex, y = 0, x = 0, and x = 2; about the x-axisSolids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 42.y ex, y ex, x 0, and x ln 4; about the x-axis17-44. Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
43. y − ln x, y − ln x2, and y − ln 8; about the y-axis
Solids of revolution Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. 44.y = ex, y = 0, x = 0, and x = p 0; about the x-axis (Is the volume bounded as p ?)Which is greater? For the following regions R, determine which is greaterthe volume of the solid generated when R is revolved about the x-axis or about the y-axis. 45.R is bounded by y = 2x, the x-axis, and x = 5.Which is greater? For the following regions R, determine which is greaterthe volume of the solid generated when R is revolved about the x-axis or about the y-axis. 46.R is bounded by y=42x, x-axis or about the y-axis.Which is greater? For the following regions R, determine which is greaterthe volume of the solid generated when R is revolved about the x-axis or about the y-axis. 47.R is bounded by y=1x3, the x-axis or about the y-axis.Which is greater? For the following regions R, determine which is greaterthe volume of the solid generated when R is revolved about the x-axis or about the y-axis. 48.R is bounded by y=x2 and y=8x.Revolution about other axes Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line. 49.y=1x,y=1, and x = 1; about y = 1Revolution about other axes Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line. 50.y=ex/2+2,y=2,x=0 and x = 1; about y = 2Revolution about other axes Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line. 51.x=2secy,x=2,y=3, and y = 0; about x = 2Revolution about other axes Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line. 52.y=1x,x=1, and y = 1; about x = 1Revolution about other axes Find the volume of the solid generated in the following situations. 45. The region R bounded by the graphs of x = 0, y=x, and y = 1 is revolved around the line y = 1.Revolution about other axes Find the volume of the solid generated in the following situations. 46. The region R bounded by the graphs of x = 0, y=x, and y = 2 is revolved around the line x = 4.Revolution about other axes Find the volume of the solid generated in the following situations. 47. The region R bounded by the graph of y = 2 sin x and the x-axis on [0, ] is revolved about the line y = 2.Revolution about other axes Find the volume of the solid generated in the following situations. 48. The region R bounded by the graph of y = ln x and the y-axis on the interval 0 y 1 is revolved about the line x = 1.Revolution about other axes Find the volume of the solid generated in the following situations. 49. The region R bounded by the graphs of y = sin x and y = 1 sin x on [6,56] is revolved about the line y = 1.Revolution about other axes Find the volume of the solid generated in the following situations. 50. The region R in the first quadrant bounded by the graphs of y = x and y=1+x2 is revolved about the line y = 3.Revolution about other axes Find the volume of the solid generated in the following situations. 51. The region R in the first quadrant bounded by the graphs of y = 2 x and y = 2 2x is revolved about the line x = 3.60EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A pyramid is a solid of revolution. b. The volume of a hemisphere can be computed using the disk method. c. Let R1 be the region bounded by y = cos x and the x-axis on [/2, /2]. Let R2 be the region bounded by y = sin x and the x-axis on [0, ]. The volumes of the solids generated when R1 and R2 are revolved about the x-axis are equal.62EFermats volume calculation (1636) Let R be the region bounded by the curve y=x+a (with a 0), the y-axis, and the x-axis. Let S be the solid generated by rotating R about the y-axis. Let T be the inscribed cone that has the same circular base as S and height a. Show that volume(S)/volume(T)=85.Solid from a piecewise function Let f(x)={xif0x22x2if2x52x+18if5x6. Find the volume of the solid formed when the region bounded by the graph of f, the x-axis, and the line x = 6 is revolved about the x-axis.65E66EEstimating volume Suppose the region bounded by the curve y=f(x) from x = 0 to x = 4 (see figure) is revolved about the x-axis to form a solid of revolution. Use left, right, and midpoint Riemann sums with n = 4 subintervals of equal length, to estimate the volume of the solid of revolution.Volume of a wooden object A solid wooden object turned on a lathe has a length of 50 cm and diameters (measured in cm) shown in the figure. (A lathe is a tool that spins and cuts a block of wood so that it has circular cross sections.) Use left Riemann sums with uniformly spaced grid points to estimate the volume of the object.Cylinder, cone, hemisphere A right circular cylinder with height R and radius R has a volume of VC = R3 (height = radius). a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC. b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.Water in a bowl A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0 h 8. Find the volume of water in the bowl as a function of h. (Check the special cases h = 0 and h = 8.)A torus (doughnut) Find the volume of the torus formed when the circle of radius 2 centered at (3, 0) is revolved about the y-axis. Use geometry to evaluate the integral.Which is greater? Let R be the region bounded by y = x2 and y=x. Use integration to determine which is greater, the volume of the solid generated when R is revolved about the x-axis or about the line y = 1.Cavalieri’s principle Cavalieri’s principle states that if two solids with equal altitudes have the same cross-sectional areas at every height, then they have equal volumes (see figure).
Use the general slicing method to justify Cavalieri’s principle.
Use Cavalieri’s principle to find the volume of a circular cylinder of radius r and height h whose axis is at an angle of to the base (see figure).
74EThe triangle bounded by the x-axis, the line y = 2x, and the line x = 1 is revolved about the y-axis. Give an integral that equals the volume of the resulting solid using the shell method.Write the volume integral in Example 4b in the case that R is revolved about the line y = 5. Example 4 Revolving about other Lines Let R be the region bounded by the curve y=x, the line y = 1, and the y-axis (Figure 6.49a). b.Use the disk/washer method to find the volume of the solid generated when R is revolved about the line y = 1 (Figure 6.49).Suppose the region in Example 5 is revolved about the y-axis. Which method (washer or shell) leads to an easier integral? Example 5 Volume by which Method? The region R is bounded by the graphs of f(x) = 2x x2 and g(x) = x on the interval [0, 1] (Figure 6.52). Use the washer method and the shell method to find the volume of the solid formed when R is revolved about the x-axis.Assume f and g are continuous with f(x) g(x) on [a, b]. The region bounded by the graphs of f and g and the lines x = a and x = b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.Fill in the blanks: A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to _____ or using the shell method and integrating with respect to _____.Fill in the blanks: A region R is revolved about the x-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to _____ or using the shell method and integrating with respect to _____.Look again at the region R in Figure 6.38 (p 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.
Figure 6.38
Let R be the region in the first quadrant bounded above by the curve y = 2 x2 and bounded below by the line y = x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis. a. What is the radius of a cylindrical shell at a point x in [0, 2]? b. What is the height of a cylindrical shell at a point x in [0, 2]? c. Write an integral for the volume of the solid using the shell method.Let R be the region bounded by the curves y=2x,y=2, and x = 4 in the first quadrant. 6.Suppose the shell method is used to determine the volume of the solid generated by revolving R about the x-axis. a. What is the radius of a cylindrical shell at a point y in [0, 2]? b. What is the height of a cylindrical shell at a point y in [0, 2]? c. Write an integral for the volume of the solid using the shell method.Let R be the region bounded by the curves y=2x,y=2, and x = 4 in the first quadrant. 7.Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line y = 2. a. What is the radius of a cylindrical shell at a point y in [0, 2]? b. What is the height of a cylindrical shell at a point y in [0, 2]? c. Write an integral for the volume of the solid using the shell method.Let R be the region bounded by the curves y=2x,y=2, and x = 4 in the first quadrant. 8.Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x = 4. a. What is the radius of a cylindrical shell at a point x in [0, 4]? b. What is the height of a cylindrical shell at a point x in [0, 4]? c. Write an integral for the volume of the solid using the shell method.Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 5. y = x x2, y = 0Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 7. y = (1 + x2)1, y = 0, x = 0, and x = 2Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 9. y = 3x, y = 3, and x = 0 (Use integration and check your answer using the volume formula for a cone.)Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 6. y = x2 + 4x + 2, y = x2 6x + 10Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 15. y=x, y = 0, and x = 4Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 14.y=x,y=2x and y = 0; about the x-axisShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 17. y = 4 x, y = 2, and x = 0Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 18. x=4y+y3,x=13, and y = 1Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 13. y = cos x2, y = 0, for 0x/2Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 18.y=6x,y=0,x=2, and x = 4; about the y-axisShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 10. y = 1 x2, x = 0, and y = 0, in the first quadrantShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 12. y=x, y = 0, and x = 1Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 21. x = y2, x = 0, and y = 3Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 22. y = x3, y = 1, and x = 0Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 19. y = x, y = 2 x, and y = 0Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. 14. y=42x2, y = 0, and x = 0, in the first quadrantShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 25.y=1(x2+1)2,y=0,x=1, and x = 2; about the y-axisShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 20. x = y2, x = 4, and y = 0Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 23. y = 2x3/2, y = 2, y = 16, and x = 0Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 24. y=sin1x,y=/2, and x = 0Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 29.y=exx,y=0,x=1, and x = 2; about the y-axisShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 30.y=lnxx2,y=0, and x = 3; about the y-axisShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 25. y=cos1x, in the first quadrantShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis. 26. y=502x2, in the first quadrantShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 33.y=x3x81,y=1; about the y-axisShell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 34.y2=lnx,y2=lnx3, and y = 2; about the x-axisWashers vs. shells Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply. 41. y = x, y = x1/3 in the first quadrant: revolved about the x-axis36EWashers vs. shells Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk / washer and shell methods. Check that your results agree and state which method is easier to apply. 44. y = (x 2)3 2, x = 0, and y = 25; revolved about the y-axisShell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis. 38.y=8,y=2x+2,x=0, and x = 2; about the y-axisShell method about other lines Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines. 33. x = 2Shell method about other lines Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines. 34. x = 1Shell method about other lines Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines. 35. y = 2Shell method about other lines Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines. 36. y = 2Shell method about other lines Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines. 43.y=2Shell method about other lines Let R be the region bounded by y = x2, x = 1, and y = 0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines. 44.y=2Different axes of revolution Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines. 37. y = 2Different axes of revolution Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines. 38. x = 1Different axes of revolution Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines. 39. y = 6Different axes of revolution Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x2, y = 1, and x = 0 is revolved about the following lines. 40. x = 2Volume of a sphere Let R be the region bounded by the upper half of the circle x2 + y2 = r2 and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis. a. Use the shell method to verify that the volume of a sphere of radius r is 433. b. Repeat part (a) using the disk method.Comparing American and rugby union footballs An ellipse centered at the origin is described by the equation x2a2+y2b2=1, where a and b are positive constants. If the upper half of such an ellipse is revolved about the x-axis, the resulting surface is an ellipsoid. a. Use the washer method to find the volume of an ellipsoid (in terms of a and b). Check your work using the shell method. b. Both American and rugby union footballs have the shape of ellipsoids. The maximum regulation size of a rugby union football corresponds to parameters of a = 6 in and b = 3.82 in, and the maximum regulation size of an American football corresponds to parameters of a = 5.62 in and b = 3.38 in. Find the volume of each football. c. Fill in the blank: At their maximum regulation sizes, the volume of a rugby union football has approximately _____ times the volume of an American football.A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis. a. Use the shell method to write an integral for the volume of the torus. b. Use the washer method to write an integral for the volume of the torus. c. Find the volume of the torus by evaluating one of the two integrals obtained in parts (a) and (b). (Hint: Both integrals can be evaluated without using the Fundamental Theorem of Calculus.)52EChoose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis. 53.y=xx4,y=0; about the x-axis.Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis. 54.y=xx4,y=0 about the y-axis.Choose your method Find the volume of the following solids using the method of your choice. 55. The solid formed when the region bounded by y = x2 and y = 2 x2 is revolved about the x-axisChoose your method Find the volume of the following solids using the method of your choice. 56. The solid formed when the region bounded by y = sin x and y = 1 sin x between x = /6 and x = 5/6 is revolved about the x-axisChoose your method Find the volume of the following solids using the method of your choice. 57. The solid formed when the region bounded by y = x, y = 2x + 2, x = 2, and x = 6 is revolved about the y-axis58EChoose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis. 59.y=lnx,y=lnx2, and y = 1; about the x-axis.Choose your method Find the volume of the following solids using the method of your choice. 60. The solid formed when the region bounded by y = 2, y = 2x + 2, and x = 6 is revolved about the y-axisChoose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis. 61.y=x2,y=2x, and x = 0, in the first quadrant; about the y-axisThe solid formed when the region bounded by y=x, the x-axis, and x = 4 is revolved about the x-axisExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution. b. If a region is revolved about the y-axis, then the shell method must be used. c. If a region is revolved about the x-axis, then in principle, it is possible to use the disk/washer method and integrate with respect to x or the shell method and integrate with respect to y.Shell method Use the shell method to find the volume of the following solids. 28. The solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cylinder of height 6 and radius 4Shell method Use the shell method to find the volume of the following solids. 27. A right circular cone of radius 3 and height 8Shell method Use the shell method to find the volume of the following solids. 30. The solid formed when a hole of radius 3 is drilled symmetrically through the center of a sphere of radius 6Shell method Use the shell method to find the volume of the following solids. 29. The solid formed when a hole of radius 3 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9Shell method Use the shell method to find the volume of the following solids. 32. A hole of radius r R is drilled symmetrically along the axis of a bullet. The bullet is formed by revolving the parabola y=6(1x2R2) about the y-axis, where 0 x R.69EA spherical cap by three methods Consider the cap of thickness h that has been sliced from a sphere of radius r (see figure). Verify that the volume of the cap is h2 (3r h)/3 using (a) the washer method, (b) the shell method, and (c) the general slicing method. Check for consistency among the three methods and check the special cases h = r and h = 0.Change of variables Suppose f(x) 0 for all x and 04f(x)dx=10. Let R be the region in the first quadrant bounded by the coordinate axes, y = f(x2), and x = 2. Find the volume of the solid generated by revolving R about the y-axis.Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.) a. 04(82x)2dx=208y(4y2)dy b. 02(25(x2+1)2)dx=215yy1dyVolumes without calculus Solve the following problems with and without calculus. A good picture helps. a. A cube with side length r is inscribed in a sphere, which is inscribed in a right circular cone, which is inscribed in a right circular cylinder. The side length (slant height) of the cone is equal to its diameter. What is the volume of the cylinder? b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube? c. A cylindrical hole 10 in long is drilled symmetrically through the center of a sphere. How much material is left in the sphere? (Enough information is given.)Wedge from a tree Imagine a cylindrical tree of radius a. A wedge is cut from the tree by making two cuts: one in a horizontal plane P perpendicular to the axis of the cylinder and one that makes an angle with P, intersecting P along a diameter of the tree (see figure). What is the volume of the wedge?75E76EWhat does the arc length formula give for the length of the line y = x between x = 0 and x = a, where a 0?What does the arc length formula give for the length of the line x = y between y = c and y = d, where d c? Is the result consistent with the result given by the Pythagorean theorem?Write the integral for the length of the curve x = sin y on the interval 0 y .Explain the steps required to find the length of a curve y = f(x) between x = a and x = b.Explain the steps required to find the length of a curve x = g(y) between y = c and y = d.Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval. 3. y = x3 + 2 on [2, 5]Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval. 4. y = 2 cos 3x on [, ]Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval. 5. y = e2x on [0, 2]Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval. 6. y = ln x on [1, 10]Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 7. y = 2x + 1 on [1, 5] (Use calculus.)Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 8. y = 4 3x on [3, 2] (Use calculus.)Arc lezngth calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 9. y = 8x 3 on [2, 6] (Use calculus.)Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 10. y=12(ex+ex)on[ln2,ln2]Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 11. y=13x3/2on[0,60]Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 12. y=3lnxx224on[1,6]Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 13. y=(x2+2)3/23on[0,1]Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 14. y=x3/23x1/2on[4,16]15EArc length calculations Find the arc length of the following curves on the given interval by integrating with respect to x. 16. y=23x3/212x1/2on[1,9]17EArc length calculations with respect to y Find the arc length of the following curves by integrating with respect to y. 30. x=2e2y+116e2y,for0yln22Arc length calculations with respect to y Find the arc length of the following curves by integrating with respect to y. 27. x = 2y 4, for 3 y 4 (Use calculus.)Arc length calculations with respect to y Find the arc length of the following curves by integrating with respect to y. 28. y=ln(xx21), for 1x2Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. 17. y = x2 on [l, 1]Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. 18. y = sin x on [0, ]Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. 19. y = ln x on [1,4]Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. 20. y=x33on[1,1]