Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
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. 25.x=y2, for 3 y 4Arc 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. 22. y=8x2on[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. 23. y = cos 2x 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. 24. y = 4x x2 on [0, 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. 25. y=1xon[1,10]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. 30.x=1y2+1, for 5 y 5Golden Gate cables The profile of the cables on a suspension bridge may be modeled by a parabola. The central span of the Golden Gate Bridge (see figure) is 1280 m long and 152 m high. The parabola y = 0.00037x2 gives a good fit to the shape of the cables, where |x| 640, and x and y are measured in meters. Approximate the length of the cables that stretch between the tops of the two towers.Gateway Arch The shape of the Gateway Arch in St. Louis (with a height and a base length of 630 ft) is modeled by the function y = 630 cosh (x/239.2) + 1260, where |x| 315, and x and y are measured in feet (see figure). The function cosh x is the hyperbolic cosine, defined by coshx=ex+ex2 (see Section 6.10 for more on hyperbolic functions). Estimate the length of the Gateway Arch.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. ab1+f(x)2dx=ab(1+f(x))dx. b. Assuming f is continuous on the interval [a, b], the length of the curve y = f(x) on [a, b] is the area under the curve y=1+f(x)2 on [a, b]. c. Arc length may be negative if f(x) 0 on part of the interval in question.Arc length for a line Consider the segment of the line y = mx + c on the interval [a, b]. Use the arc length formula to show that the length of the line segment is (ba)1+m2. Verify this result by computing the length of the line segment using the distance formula.Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions. a. ab1+16x4dx b. ab1+36cos22xdxFunction from arc length Find a curve that passes through the point (1, 5) and has an arc length on the interval [2, 6] given by 261+16x6dx.37E38ELengths of related curves Suppose the graph of f on the interval [a, b] has length L, where f is continuous on [a, b]. Evaluate the following integrals in terms of L. a. a/2b/21+f(2x)2dx b. a/cb/c1+f(cx)2dxifc040EA family of exponential functions a. Show that the arc length integral for the function f(x)=Aeax+14Aa2eax, where a 0 and A 0, may be integrated using methods you already know. b. Verify that the arc length of the curve y = f(x) on the interval [0, ln 2] is A(2a1)14a2A(2a1).Bernoullis parabolas Johann Bernoulli (16671748) evaluated the arc length of curves of the form y = x(2n + 1)/2n, where n is a positive integer, on the interval [0, a]. a. Write the arc length integral. b. Make the change of variables u2=1+(2n+12n)2x1/nto obtain a new integral with respect to u. c. Use the Binomial Theorem to expand this integrand and evaluate the integral. d. The case n = 1 (y = x3/2) was done in Example 1. With a = 1, compute the arc length in the cases n = 2 and n = 3. Does the arc length increase or decrease with n? e. Graph the arc length of the curves for a = 1 as a function of n.Which is greater the surface area of a cone of height 10 and rad us 20 or the surface area of a cone of height 20 and radius 10 (excluding the bases)?What is the surface area of the frustum of a cone generated when the graph of f(x) = 3x on the interval [2, 5] is revolved about the x-axis?Let f(x) = c, where c 0. What surface is generated when the graph of f on [a, b] is revolved about the x-axis? Without using calculus, what is the area of the surface?What is the area of the curved surface of a right circular cone of radius 3 and height 4?A frustum of a cone is generated by revolving the graph of y = 4x on the interval [2, 6] about the x-axis. What is the area of the surface of the frustum?Suppose f is positive and differentiable on [a, b]. The curve y = f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.Suppose g is positive and differentiable on [c, d]. The curve x = g(y) on [c, d] is revolved about the y-axis. Explain how to find the area of the surface that is generated.A surface is generated by revolving the line f(x) = 2 x, for 0 x 2, about the x-axis. Find the area of the resulting surface in the following ways. a. Using calculus b. Using geometry after first determining the shape and dimensions of the surfaceA surface is generated by revolving the line x = 3, for 0 y 8, about the y-axis. Find the area of the resulting surface in the following ways. a. Using calculus with g(y) = 3 b. Using geometry, after first determining the shape and dimensions of the surfaceComputing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. 5. y = 3x + 4 on [0, 6]Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. 6. y = 12 3x on [1, 3]Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. 7. y=8x on [9, 20]Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. 8. y = x3 on [0, l]Revolving about the y-axis Find the area of the surface generated when the given curve is revolved about the y-axis. 17. y = (3x)1/3, for 0x83Revolving about the y-axis Find the area of the surface generated when the given curve is revolved about the y-axis. 18. y=x24, for 2 x 4Computing surface areas Find the area of the surface generated when the given curve is revolved about the given axis. 13.y=1x2, for 12x12; about the x-axisComputing surface areas Find the area of the surface generated when the given curve is revolved about the given axis. 14.x=y2+6y8, 3y72; about the y-axisComputing surface areas Find the area of the surface generated when the given curve is revolved about the given axis. 15.y=4x1, for 1x4; about the y-axis (Hint: Integrate with respect to y.)Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. 10. y=4x+6 on [0, 5]Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. 11. y=14(e2x+e2x) on [2, 2]Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. 14. y=5xx2 on [1, 4]Surface area calculations Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. 22. x=12yy2, for 2 y 10; about the y-axisSurface area calculations Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. 24. y=1+1x2 between the points (1, 1) and (32,32); about the y-axisPainting surfaces A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume that x and y are measured in meters. 15. The spherical zone generated when the curve y=8xx2 on the interval [1, 7] is revolved about the x-axisPainting surfaces A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume that x and y are measured in meters. 16. The spherical zone generated when the upper portion of the circle x2 + y2 = 100 on the interval [8, 8] is revolved about the x-axisExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the curve y = f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is f(a)f(b)2f(y)1+f(y)2dy. b. If f is not one-to-one on the interval [a, b] then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined. c. Let f(x) = 12x2. The area of the surface generated when the graph of f on [4, 4] is revolved about the x-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the x-axis. d. Let f(x) = 12x2. The area of the surface generated when the graph of f on [4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.24ET 2629. Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the x-axis. b. Use a calculator or software to approximate the surface area. 27. y = cos x on [0,2]Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis. b. Use a calculator or software to approximate the surface area. 26.y=ex, for 0x1; about the y-axisSurface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the x-axis. b. Use a calculator or software to approximate the surface area. 29. y = tan x on [0,4]28E29ECones and cylinders The volume of a cone of radius r and height h is one-third the volume of a cylinder with the same radius and height. Does the surface area of a cone of radius r and height h equal one-third the surface area of a cylinder with the same radius and height? If not, find the correct relationship. Exclude the bases of the cone and cylinder.Challenging surface area calculations Find the area of the surface generated when the given curve s revolved about the given axis. 31.y=x3/2x1/23, for 1x2; about the x-axisChallenging surface area calculations Find the area of the surface generated when the given curve s revolved about the given axis. 32.y=x48+14x2, for 1x2; about the x-axisChallenging surface area calculations Find the area of the surface generated when the given curve s revolved about the given axis. 33.y=x33+14x, for 12x2; about the x-axisChallenging surface area calculations Find the area of the surface generated when the given curve s revolved about the given axis. 34.y=12ln(2x+4x21) between the points (12,0) and (1716,ln2); about the y-axisSurface area calculations Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. 23. x=4y3/2y1/212, for 1 y 4; about the y-axisSurface area of a torus When the circle x2 + (y a)2 = r2 on the interval [r, r] is revolved about the x-axis, the result is the surface of a torus, where 0 r a. Show that the surface area of the torus is S = 42ar.Zones of a sphere Suppose a sphere of radius r is sliced by two horizontal planes h units apart (see figure). Show that the surface area of the resulting zone on the sphere is 2rh, independent of the location of the cutting planes.38ESurface-area-to-volume ratio (SAV) In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly. a. What is the SAV ratio of a cube with side lengths a? b. What is the SAV ratio of a ball with radius a? c. Use the result of Exercise 34 to find the SAV ratio of an ellipsoid whose long axis has length 2a43, for a 1, and whose other two axes have half the length of the long axis. (This scaling is used so that, for a given value of a, the volumes of the ellipsoid and the ball of radius a are equal.) The volume of a general ellipsoid is V=43ABC, where the axes have lengths 2A, 2B, and 2C. d. Graph the SAV ratio of the ball of radius a 1 as a function of a (part (b)) and graph the SAV ratio of the ellipsoid described in (part (c)) on the same set of axes. Which object has the smaller SAV ratio? e. Among all ellipsoids of a fixed volume, which one would you choose for the shape of an animal if the goal is to minimize heat loss?Surface area of a frustum Show that the surface area of the frustum of a cone generated by revolving the line segment between (a, g(a)) and (b, g(b)) about the x-axis is (g(b) + g(a)), for any linear function g(x) = cx + d that is positive on the interval [a, b], where is the slant height of the frustum.Scaling surface area Let f be a nonnegative function with a continuous first derivative on [a, b] and suppose that g(x) = cf(x) and h(x) = f(cx), where c 0. When the curve y = f(x) on [a, b] is revolved about the x-axis, the area of the resulting surface is A. Evaluate the following integrals in terms of A and c. a. ab2g(x)c2+g(x)2dx b. a/cb/c2h(x)c2+h(x)2dxSurface plus cylinder Suppose f is a nonnegative function with a continuous first derivative on [a, b]. Let L equal the length of the graph of f on [a, b] and let S be the area of the surface generated by revolving the graph of f on [a, b] about the x-axis. For a positive constant C, assume the curve y = f(x) + C is revolved about the x-axis. Show that the area of the resulting surface equals the sum of S and the surface area of a right circular cylinder of radius C and height L.In Figure 6.69, suppose a = 0, b = 3, and the density of the rod in g/cm is (x)=(4x). (a) Where is the rod lightest and heaviest? (b) What is the density at the middle of the bar?A thin bar occupies the interval 0 x 2 and has a density in kg/m of (x)=(1+x2). Using the minimum value of the density, what is a lower bound for the mass of the object? Using the maximum value of the density, what is an upper bound for the mass of the object?3QC4QCIn Example 3b, the bucket occupies the interval [0, 1] and the chain occupies the interval [1, 11] (Figure 6.75b). Why is integration used to compute the work needed to lift the chain but not to compute the work needed to lift the bucket?
Example 3
Lifting a chain and bucket A ten-meter chain with density of 1.5 kg/m hangs from a platform at a construction site that is 11 meters above the ground (Figure 6.75a).
Figure 6.75
Compute the work required to lift the chain to the platform.
Several packages of nails are placed in a one-meter-tall bucket that rests on the ground; the mass of the bucket and nails together is 15 kg, and the chain is attached to the bucket (Figure 6.75b). How much work is required to lift the bucket to the platform?
6QCIn Example 4, how would the integral change if the outflow pipe were at the top of the tank? Example 4 Pumping Water How much work is needed to pump all the water out of a cylindrical tank with a height of 10 m and a radius of 5 m? The water is pumped to an outflow pipe 15 m above the bottom of the tank.Suppose a 1-m cylindrical bar has a constant density of 1 g/cm for its left half and a constant density 2 g/cm for its right half. What is its mass?Explain how to find the mass of a one-dimensional object with a variable density .How much work is required to move an object from x = 0 to x = 5 (measured in meters) in the presence of a constant force of 5 N acting along the x-axis?Why is integration used to find the work done by a variable force?Why is integration used to find the work required to pump water out of a tank?Why is integration used to find the total force on the face of a dam?What is the pressure on a horizontal surface with an area of 2 m2 that is 4 m underwater?Explain why you integrate in the vertical direction (parallel to the acceleration due to gravity) rather than the horizontal direction to find the force on the face of a dam.Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then 01025g(15y)dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.) 9.The work required to empty the top half of the tankConsider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then 01025g(15y)dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.) 10.The work required to empty the tank if it is half fullConsider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then 01025g(15y)dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.) 11.The work required to empty the tank through an outflow pipe at the top of the tankConsider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then 01025g(15y)dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.) 12.The work required to empty the tank if the water in the tank is only 3 m deep and the outflow pipe is at the top of the tankMass of one-dimensional objects Find the mass of the following thin bars with the given density function. 9. (x) = 1 + sin x; for 0 xMass of one-dimensional objects Find the mass of the following thin bars with the given density function. 10. (x) = 1 + x3; for 0 x 1Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. 11. (x) = 2 x/2; for 0 x 2Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. 12. (x) = 5e2x; for 0 x 4Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. 13. (x)=x2x2; for 0 x 1Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. 14. (x)={1if0x22if2x3Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. 15. (x)={1if0x21+xif2x4Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. 16. (x)={x2if0x1x(2x)if1x2Work from force How much work is required to move an object from x = 0 to x = 3 (measured in meters) in the presence of a force (in N) given by F(x) = 2x acting along the x-axis?Work from force How much work is required to move an object from x = 1 to x = 3 (measured in meters) in the presence of a force (in N) given by F(x) = 2/x2 acting along the x-axis?Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position. a. Assuming the spring obeys Hookes law, find the spring constant k. b. How much work is required to compress the spring 0.4 m from its equilibrium position? c. How much work is required to stretch the spring 0.3 m from its equilibrium position? d. How much additional work is required to stretch the spring 0.2 m if it has already been stretched 0.2 m from its equilibrium position?Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position. a. Assuming the spring obeys Hookes law, find the spring constant k. b. How much work is required to compress the spring 0.2 m from its equilibrium position? c. How much additional work is required to stretch the spring 0.3 m if it has already been stretched 0.25 m from its equilibrium position?Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N. a. How much work is done in stretching the spring 1.5 m from its equilibrium position? b. How much work is done in compressing the spring 0.5 m from its equilibrium position?Shock absorber A heavy-duty shock absorber is compressed 2 cm from its equilibrium position by a mass of 500 kg. How much work is required to compress the shock absorber 4 cm from its equilibrium position? (A mass of 500 kg exerts a force (in newtons) of 500 g, where g 9.8 m/s2.)Calculating work for different springs Calculate the work required to stretch the following springs 0.5 m from their equilibrium positions. Assume Hookes law is obeyed. a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position b. A spring that requires 50 J of work to be stretched 0.2 m from its equilibrium positionCalculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hookes law is obeyed. a. A spring that requires a force of 50 N to be stretched 0.1 m from its equilibrium position b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium positionCalculating work for different springs Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hookes law is obeyed. a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position b. A spring that requires a force of 250 N to be stretched 0.5 m from its equilibrium positionWork function A spring has a restoring force given by F(x) = 25x. Let W(x) be the work required to stretch the spring from its equilibrium position (x = 0) to a variable distance x. Find and graph the work function. Compare the work required to stretch the spring x units from equilibrium to the work required to compress the spring x units from equilibrium.Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5 kg/m. a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?Coiling a rope A 60-m-long, 9.4-mm-diameter rope hangs free from a ledge. The density of the rope is 55 g /m. How much work is needed to pull the entire rope to the ledge?Winding part of a chain A 20-m-long, 50-kg chain hangs vertically from a cylinder attached to a winch. How much work is needed to wind the upper half of the chain onto the winch?Leaky Bucket A 1-kg bucket resting on the ground contains 3 kg of water. How much work is required to raise the bucket vertically a distance of 10 m if water leaks out of the bucket at a constant rate of 15kg/m? Assume the weight of the rope used to raise the bucket is negligible. (Hint: Use the definition of work, W=abF(y)dy,, where F is the variable force required to lift an object along a vertical line from y = a to y = b.)Emptying a swimming pool A swimming pool has the shape of a box with a base that measures 25 m by 15 m and a uniform depth of 2.5 m. How much work is required to pump the water out of the pool when it is full?Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2 m (see figure). a. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank? b. Is it true that it takes half as much work to pump the water out of the tank when it is half full as when it is full? Explain.Emptying a half-full cylindrical tank Suppose the water tank in Exercise 36 is half full of water. Determine the work required to empty the tank by pumping the water to a level 2 m above the top of the tank.Emptying a partially filled swimming pool If the water in the swimming pool in Exercise 35 is 2 m deep, then how much work is required to pump all the water to a level 3 m above the bottom of the pool?Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure). a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank? b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.Upper and lower half A cylinder with height 8 m and radius 3 m is filled with water and must be emptied through an outlet pipe 2 m above the top of the cylinder. a. Compute the work required to empty the water in the top half of the tank. b. Compute the work required to empty the (equal amount of) water in the lower half of the tank. c. Interpret the results of parts (a) and (b).Filling a spherical tank A spherical water tank with an inner radius of 8 m has its lowest point 2 m above the ground. It is filed by a pipe that feeds the tank at its lowest point (see figure). Neglecting the volume of the inflow pipe, how much work is required to fill the tank if it is initially empty?Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure). a. How much work is required to pump the water out of the trough when it is full? b. If the length is doubled, is the required work doubled? Explain. c. If the radius is doubled, is the required work doubled? Explain.Emptying a water trough A cattle trough has a trapezoidal cross section with a height of 1 m and horizontal sides of length 12 m and 1 m. Assume the length of the trough is 10 m (see figure). a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full? b. If the length is doubled, is the required work doubled? Explain.Pumping water Suppose the tank in Example 5 is full of water (rather than half full of gasoline). Determine the work required to pump all the water to an outlet pipe 15 m above the bottom of the tank.Emptying a conical tank An inverted cone is 2 m high and has a base radius of 12 m. If the tank is full, how much work is required to pump the water to a level 1 m above the top of the tank?Force on dams The following figures show the shape and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam. 38.Force on dams The following figures show the shape and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam. 39.Force on dams The following figures show the shape and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam. 40.Force on dams The following figures show the shape and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam. 41.Parabolic dam The lower edge of a dam is defined by the parabola y = x2/16 (see figure). Use a coordinate system with y = 0 at the bottom of the dam to determine the total force on the dam. Lengths are measured in meters. Assume the water level is at the top of the dam.51EForce on a window A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. 46. The window is a square, 0.5 m on a side, with the lower edge of the window on the bottom of the pool.Force on a window A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. 47. The window is a square, 0.5 m on a side, with the lower edge of the window 1 m from the bottom of the pool.Force on a window A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. 48. The window is a circle, with a radius of 0.5 m, tangent to the bottom of the pool.Force on a building A large building shaped like a box is 50 m high with a face that is 80 m wide. A strong wind blows directly at the face of the building, exerting a pressure of 150 N/m2 at the ground and increasing with height according to P(y) = 150 + 2y, where y is the height above the ground. Calculate the total force on the building, which is a measure of the resistance that must be included in the design of the building.Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.76 if the tank is full of gasoline. The density of gasoline is = 737 kg/m3. Figure 6.76Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The mass of a thin wire is the length of the wire times its average density over its length. b. The work required to stretch a linear spring (that obeys Hookes law) 100 cm from equilibrium is the same as the work required to compress it 100 cm from equilibrium. c. The work required to lift a 10-kg object vertically 10 m is the same as the work required to lift a 20-kg object vertically 5 m. d. The total force on a 10-ft2 region on the (horizontal) floor of a pool is the same as the total force on a 10-ft2 region on a (vertical) wall of the pool.58EA nonlinear spring Hookes law is applicable to idealized (linear) springs that are not stretched or compressed too far. Consider a nonlinear spring whose restoring force is given by F(x) = 16x 0.1x3, for |x| 7. a. Graph the restoring force and interpret it. b. How much work is done in stretching the spring from its equilibrium position (x = 0) to x = 1.5? c. How much work is done in compressing the spring from its equilibrium position (x = 0) to x = 2?60ELeaky cement bucket A 350 kg-bucket containing 4650 kg of cement is resting on the ground when a crane begins lifting it at a constant rate of 5 m/min. As the crane raises the bucket, cement leaks out of the bucket at a constant rate of 100 kg/min. How much work is required to lift the bucket a distance of 30 m if we ignore the weight of the crane cable attached to the bucket?Emptying a real swimming pool A swimming pool is 20 m long and 10 m wide, with a bottom that slopes uniformly from a depth of 1 m at one end to a depth of 2 m at the other end (see figure). Assuming the pool is full, how much work is required to pump the water to a level 0.2 m above the top of the pool?Drinking juice A glass has circular cross sections that taper (linearly) from a radius of 5 cm at the top of the glass to a radius of 4 cm at the bottom. The glass is 15 cm high and full of orange juice. How much work is required to drink all the juice through a straw if your mouth is 5 cm above the top of the glass? Assume the density of orange juice equals the density of water.Lifting a pendulum A body of mass m is suspended by a rod of length L that pivots without friction (see figure). The mass is slowly lifted along a circular arc to a height h. a. Assuming that the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is F = mg sin . b. Noting that an element of length along the path of the pendulum is ds = L d, evaluate an integral in to show that the work done in lilting the mass to a height h is mgh.Critical depth A large tank has a plastic window on one wall that is designed to withstand a force of 90,000 N. The square window is 2 m on a side, and its lower edge is 1 m from the bottom of the tank. a. If the tank is filled to a depth of 4 m, will the window withstand the resulting force? b. What is the maximum depth to which the tank can be filled without the window failing?66E67E68EWork in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm/(x + R)2, where G = 6.7 1011 Nm2/kg2 is the gravitational constant, M = 6 1024 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378 106 m is the radius of Earth, and x 0 is the distance above the surface of Earth (in meters). a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earths surface)? b. Find the work required to launch the rocket to a height of x kilometers, for x 0. c. How much work is required to reach outer space (x )? d. Equate the work in part (c) to the initial kinetic energy of the rocket, 12mv2, to compute the escape velocity of the rocket.Buoyancy Archimedes principle says that the buoyant force exerted on an object that is (partially or totally) submerged in water is equal to the weight of the water displaced by the object (see figure). Let w = 1 g/cm3 = 1000 kg/m3 be the density of water and let be the density of an object in water. Let f = /w. If 0 f 1, then the object floats with a fraction f of its volume submerged; if f 1, then the object sinks. Consider a cubical box with sides 2 m long floating in water with one-half of its volume submerged ( = w/2). Find the force required to fully submerge the box (so its top surface is at the water level). (See the Guided Project Buoyancy and Archimedes Principle for further explorations of buoyancy problems.)Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A region R is revolved about the y-axis to generate a solid S. To find the volume of S, you could either use the disk/washer method and integrate with respect to y or use the shell method and integrate with respect to x. b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position. c. If water flows into a tank at a constant rate (for example 6 gal/min), the volume of water in the tank increases according to a linear function of time.2REDisplacement, distance, and position Consider an object moving along a line with the following velocities and initial positions. Assume time t is measured in seconds and velocities have units of m/s. a.Over the given interval, determine when the object is moving in the positive direction and when it is moving in the negative direction. b.Find the displacement over the given interval. c.Find the distance traveled over the given interval. d.Determine the position function s(t) using the Fundamental Theorem of Calculus (Theorem 6.1). Check your answer by finding the position function using the antiderivative method. 3.v(t)=12t230t+12, for 0t3;s(0)=1Displacement from velocity The velocity of an object moving along a line is given by v(t) = 20 cos t (in ft/s). What is the displacement of the object after 1.5 s?Position, displacement, and distance A projectile is launched vertically from the ground at t = 0, and its velocity in flight (in m/s) is given by v(t) = 20 10t. Find the position, displacement, and distance traveled after t seconds, for 0 t 4.Deceleration At t = 0, a car begins decelerating from a velocity of 80 ft/s at a constant rate of 5 ft/s2. Find its position function assuming s(0) = 0.An oscillator The acceleration of an object moving along a line is given by a(t)=2sin(t4). The initial velocity and position are v(0)=8 and s(0) = 0. a. Find the velocity and position for t 0. b. What are the minimum and maximum values of s? c. Find the average velocity and average position over the interval [0, 8].A race Starting at the same point on a straight road, Anna and Benny begin running with velocities (in mi/hr) given by vA(t) = 2t + 1 and vB(t) = 4 1, respectively. a. Graph the velocity functions, for 0 t 4. b. If the runners run for 1 hr, who runs farther? Interpret your conclusion geometrically using the graph in part (a). c. If the runners run for 6 mi, who wins the race? Interpret your conclusion geometrically using the graph in part (a).Fuel consumption A small plane in flight consumes fuel at a rate (in gal/min) given by R(t)={4t1/3if0t8(take-off)2ift8(cruising). a. Find a function R that gives the total fuel consumed, for 0 t 8. b. Find a function R that gives the total fuel consumed, for t 0. c. If the fuel tank capacity is 150 gal, when does the fuel run out?Variable flow rate Water flows out of a tank at a rate (in m3/hr) given by V(t) = 15/(t + 1). If the tank initially holds 75 m3 of water, when will the tank be empty?Decreasing velocity A projectile is fired upward, and its velocity in m/s is given by v(t) = 200et/10, for t 0. a. Graph the velocity function for t 0. b. When does the velocity reach 50 m/s? c. Find and graph the position function for the projectile for t 0, assuming s(0) = 0. d. Given unlimited time, can the projectile travel 2500 m? If so, at what time does the distance traveled equal 2500 m?Decreasing velocity A projectile is fired upward, and its velocity (in m/s) is given by v(t)=200t+1, for t 0. a. Graph the velocity function for t 0. b. Find and graph the position function for the projectile, for t 0, assuming s(0) = 0. c. Given unlimited time, can the projectile travel 2500 m? If so, at what time does the distance traveled equal 2500 m?An exponential bike ride Tom and Sue took a bike ride, both starting at the same time and position. Tom started riding at 20 mi/hr, and his velocity decreased according to the function v(t) = 20e2t for t 0. Sue started riding at 15 mi/hr, and her velocity decreased according to the function u(t) = 15e1 for t 0. a. Find and graph the position functions of Tom and Sue. b. Find the times at which the riders had the same position at the same time. c. Who ultimately took the lead and remained in the lead?Areas of regions Determine the area of the given region. 14.Areas of regions Determine the area of the given region. 15.Areas of regions Determine the area of the given region. 16.Areas of regions Determine the area of the given region. 17.The region bounded by y=lnx,y=1, and x = 118REAreas of regions Use any method to find the area of the region described. 13. The region in the first quadrant bounded by y = 4x and y=x25x2Areas of regions Determine the area of the given region. 20.The region bounded by y=63x2 and y=6x3Areas of regions Use any method to find the area of the region described. 17. The region bounded by y = x2, y = 2x2 4x, and y = 0Areas of regions Use any method to find the area of the region described. 18. The region in the first quadrant bounded by the curve x+y=1Areas of regions Use any method to find the area of the region described. 19. The region in the first quadrant bounded by y = x/6 and y = 1 |x/2 1|24REAreas of regions Determine the area of the given region. 25.The region bounded by y=2x26x+5 and y = 1Multiple regions Determine the area of the region bounded by the curves x = y2 and x = (2 y2)2 (see figure).Multiple regions The regions R1, R2, and R3 (see figure) are formed by the graphs of y2x,y3x, and x 3. 27.Find the area of each of the regions R1, R2, and R3.28REMultiple regions The regions R1, R2, and R3 (see figure) are formed by the graphs of y2x,y3x, and x 3. 29.Find the volume of the solid obtained by revolving region R1 about teh y-axis.30REMultiple regions The regions R1, R2, and R3 (see figure) are formed by the graphs of y2x,y3x, and x 3. 31.Find the volume of the solid obtained by revolving region R2 about the x-axis.Multiple regions The regions R1, R2, and R3 (see figure) are formed by the graphs of y2x,y3x, and x 3. 32.Use the disk method to find an integral, or sum of integrals, that equals the volume of the solid obtained by revolving region R3 about the line x = 3. Do not evaluate the integral.Multiple regions The regions R1, R2, and R3 (see figure) are formed by the graphs of y2x,y3x, and x 3. 33.Use the shell method to find an integral, or sun of integrals, that equals the volume of the solid obtained by revolving region R3 about the line x = 3. Do not evaluate the integral.Area and volume The region R is bounded by the curves x = y2 + 2, y = x 4, and y = 0 (see figure). a. Write a single integral that gives the area of R. b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis. c. Write a single integral that gives the volume of the solid generated when R is revolved about the y-axis. d. Suppose S is a solid whose base is R and whose cross sections perpendicular to R and parallel to the x-axis are semicircles. Write a single integral that gives the volume of S.Area and volume Let R be the region in the first quadrant bounded by the graph of f(x)={1if0x12xif1x4. 35.Find the area of the region R.Area and volume Let R be the region in the first quadrant bounded by the graph of f(x)={1if0x12xif1x4. 36.Find the volume of the solid generated when R is revolved about the x-axis.Area and volume Let R be the region in the first quadrant bounded by the graph of f(x)={1if0x12xif1x4. 37. Find the volume of the solid generated when R is revolved about the y-axis.Area and volume Let R be the region in the first quadrant bounded by the graph of f(x)={1if0x12xif1x4. 38.Find the area of the surface generated when the curve y=f(x), for 0x4, is revolved about the x-axis.Find the area of the shaded regions R1 and R2 shown in the figure.40RE41RETwo methods The region R in the first quadrant bounded by the parabola y = 4 x2 and the coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways. a. Apply the disk method and integrate with respect to y. b. Apply the shell method and integrate with respect to x.Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 23. What is the volume of the solid whose base is the region in the first quadrant bounded by y=x, y = 2 x, and the x-axis, and whose cross sections perpendicular to the base and parallel to the y-axis are squares?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 24. What is the volume of the solid whose base is the region in the first quadrant bounded by y=x, y = 2 x, and the x-axis, and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 25. What is the volume of the solid whose base is the region in the first quadrant bounded by y=x, y = 2 x, and the y-axis, and whose cross sections perpendicular to the base and parallel to the x-axis are square?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 26. The region bounded by the curves y = x2 + 2x + 2 and y = 2x2 4x + 2 is revolved about the x-axis. What is the volume of the solid that is generated?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 27. The region bounded by the curves y=1+x, y=1x, and the line x = 1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 28. The region bounded by the curves y = 2ex, y = ex, and the y-axis is revolved about the x-axis. What is the volume of the solid that is generated?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 29. The region bounded by the graphs of x = 0, x=lny, and x=2lnyin the first quadrant is revolved about the y-axis. What is the volume of the resulting solid?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 30. The region bounded by the curves y = sec x and y = 2, for 0x3, is revolved around the x-axis. What is the volume of the solid that is generated?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 31. The region bounded by y = (1 x2)1/2 and the x-axis over the interval [0,3/2] is revolved around the y-axis. What is the volume of the solid that is generated?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 32. The region bounded by the graph of y = 4 x2 and the x-axis on the interval [2, 2] is revolved about the line x = 2. What is the volume of the solid that is generated?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 33. The region bounded by the graphs of y = (x 2)2 and y = 4 is revolved about the line y = 4. What is the volume of the resulting solid?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 34. The region bounded by the graphs of y = 6x and y = x2 + 5 is revolved about the line y = 1 and the line x = 1. Find the volumes of the resulting solids. Which one is greater?Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. 35. The region bounded by the graphs of y = 2x, y = 6 x, and y = 0 is revolved about the line y = 2 and the line x = 2. Find the volumes of the resulting solids. Which one is greater?Comparing volumes Let R be the region bounded by y = 1/xp and the x-axis on the interval [1, a], where p 0 and a 1 (see figure). Let Vx, and Vy be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively. a. With a = 2 and p = 1, which is greater, Vx or Vy? b. With a = 4 and p = 3, which is greater, Vx or Vy? c. Find a general expression for Vx in terms of a and p. Note that p=12 is a special case. What is Vx when p=12? d. Find a general expression for Vy in terms of a and p. Note that p = 2 is a special case. What is Vy when p = 2? e. Explain how parts (c) and (d) demonstrate that limh0ah1h=lna. f. Find any values of a and p for which Vx Vy.Comparing volumes Let R be the region bounced by the graph of f(x)=cx(1x) and the x-axis on [0, 1]. Find the positive value of c such that the volume of the solid generated by revolving R about the x-axis equals the volume of the solid generated by revolving R about the y-axisArc length Find the length of the following curves. 40. y = 2x + 4 on the interval [2, 2] (Use calculus.)Arc length Find the length of the following curves. 59.y=ln(x+x21) on [2,5]Arc length Find the length of the following curves. 42. y = x3/6 + l/(2x) on the interval [1, 2]Arc length Find the length of the following curves. 43. y = x1/2 x3/2/3 on the interval [1, 3]Arc length by calculator Write and simplify the integral that gives the arc length of the following curves on the given interval. Then use technology to approximate the integral. 62.y=sinx on [0, 1]Arc length by calculator Write and simplify the integral that gives the arc length of the following curves on the given interval. Then use technology to approximate the integral. 63.y=(x+1)2 on [2,4]Arc length by calculator Write and simplify the integral that gives the arc length of the following curves on the given interval. Then use technology to approximate the integral. 64.y=x33+x22 on [0, 2]Arc length by calculator Write and simplify the integral that gives the arc length of the following curves on the given interval. Then use technology to approximate the integral.
Surface area and volume Let f(x)=13x3 and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2]. a. Find the area of the surface generated when the graph of f on [0, 2] is revolved about the x-axis. b. Find the volume of the solid generated when R is revolved about the y-axis. c. Find the volume of the solid generated when R is revolved about the x-axis.Surface area and volume Let f(x)=3xx2 and let R be the region bounded by the graph of f and the x-axis on the interval [0, 3]. a. Find the area of the surface generated when the graph of f on [0, 3] is revolved about the x-axis. b. Find the volume of the solid generated when R is revolved about the x-axis.Surface area of a cone Find the surface area of a cone (excluding the base) with radius 4 and height 8 using integration and a surface area integral.Surface area and more Let f(x)=x42+116x2 and let R be the region bounded by the graph of f and the x-axis on the interval [1, 2]. a. Find the area of the surface generated when the graph of f on [1, 2] is revolved about the x-axis. b. Find the length of the curve y = f(x) on [1, 2]. c. Find the volume of the solid generated when R is revolved about the y-axis. d. Find the volume of the solid generated when R is revolved about the x-axis.Variable density in one dimension Find the mass of the following thin bars. 50. A bar on the interval 0 x 9 with a density (in g/cm) given by (x)=3+2xVariable density in one dimension Find the mass of the following thin bars. 51. A 3-m bar with a density (in g/m) of (x) = 150ex/3, for 0 x 3Variable density in one dimension Find the mass of the following thin bars. 52. A bar on the interval 0 x 6 with a density (x)={1if0x22if2x44if4x6.Spring work a. It lakes 50 J of work to stretch a spring 0.2 m from its equilibrium position. How much work is needed to stretch it an additional 0.5 m? b. It takes 50 N of force to stretch a spring 0.2 m from its equilibrium position. How much work is needed to stretch it an additional 0.5 m?Leaky bucket A 1-kg bucket resting on the ground init ally contains 6 kg of water. The bucket is lifted from the ground at a constant rate, and while it is rising, water leaks out of the bucket at a constant rate of 14kg/m. How much work s required to lift the bucket a vertical distance of 8 m? Assume the weight of the rope used to raise the bucket is negligible.Lifting problem A 10-m, 20-kg chain hangs vertically from a cylinder attached to a winch. a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the upper 4 m of the chain onto the cylinder using the winch?Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?Pumping water A water tank has the shape of a box that is 2 m wide, 4 m long, and 6 m high. a. If the tank is full, how much work is required to pump the water to the level of the top of the tank? b. If the water in the tank is 2 m deep, how much work is required to pump the water to a level of 1 m above the top of the tank?Pumping water A cylindrical water tank has a height of 6 m and a radius of 4 m. How much work is required to empty the full tank by pumping the water to an outflow pipe at the top of the tank?Pumping water A water tank that is full of water has the shape of an inverted cone with a height of 6 m and a radius of 4 m (see figure). Assume the water is pumped out to the level of the top of the tank. a. How much work is required to pump out the water? b. Which requires more work, pumping out the top 3 m of water or the bottom 3 m of water?Pumping water A water tank that has the shape of a sphere with a radius of 2 m is half full of water. How much work is required to pump the water to the level of the top of the tank?Pumping water A tank has the shape of the surface obtained by revolving the curve y = x2, for 0 ≤ x ≤ 3, about the y-axis. (Assume x and y are in meters.) If the tank is full of water, how much work is required to pump the water to an outflow pipe 1 m above the top of the tank?
Fluid Forces Suppose the Mowing plates are placed on a vertical wall so that the top of the plate is 2 m below the surface of a pool that is filled with water. Compute the force on each plate. 82.A square plate with side length of 1 m, oriented so that the top and bottom sides of the square are horizontalFluid Forces Suppose the Mowing plates are placed on a vertical wall so that the top of the plate is 2 m below the surface of a pool that is filled with water. Compute the force on each plate. 83.A plate shaped like an isosceles triangle with a height of 1 m and a base of length 12 m (see figure)Fluid Forces Suppose the Mowing plates are placed on a vertical wall so that the top of the plate is 2 m below the surface of a pool that is filled with water. Compute the force on each plate. 84.A circular plate with a radius of 2mForce on a dam Find the total force on the face of a semicircular dam with a radius of 20 m when its reservoir is full of water. The diameter of the semicircle is the top of the dam.Equal area property for parabolas Let f(x) = ax2 + bx + c be an arbitrary quadratic function and choose two points x = p and x= q. Let L1 be the line tangent to the graph of f at the point (p, f(p)) and let L2 be the line tangent to the graph at the point (q, f(q)). Let x = s be the vertical line through the intersection point of L1 and L2. Finally, let R1, be the region bounded by y =f(x), L1, and the vertical line x = s, and let R2 be the region bounded by y = f(x), L2, and the vertical line x = s. Prove that the area of R1 equals the area of R2.What is the domain of ln |x|?Simplify e ln 2x, ln (e2x), e2 ln x, and ln (2ex)What is the slope of the curve y = ex at x= ln 2? What is the area of the region bounded by the graph of y = ex and the x-axis between x = 0 and x = ln 2?Verify that the derivative and integral results for a general base b reduce to the expected results when b = e.1E2EEvaluate 4xdx.What is the inverse function of ln x, and what are its domain and range?Express 3x, x, and xsin x using the base e.Evaluate ddx(3x).Derivatives Evaluate the following derivatives ddx(xlnx3)Derivatives with ln x Evaluate the following derivatives. 8. ddx(ln(lnx))Derivatives with ln x Evaluate the following derivatives. 9. ddx(sin(lnx))Derivatives with ln x Evaluate the following derivatives. 10. ddx(ln(cos2x))Derivatives with ln x Evaluate the following derivatives. 11. ddx((ln2x)5)Derivatives with ln x Evaluate the following derivatives. 12. ddx(ln3(3x2+2))Derivatives Evaluate the derivatives of the following functions. 33. f(x) = (2x)4xDerivatives Evaluate the derivatives of the following functions. 34. f(x) = xDerivatives Evaluate the derivatives of the following functions. 35. h(x)=2(x2)Derivatives Evaluate the derivatives of the following functions. 36. h(t)=(sint)tDerivatives Evaluate the derivatives of the following functions. 37. H(x) = (x + l)2xDerivatives Evaluate the derivatives of the following functions. 38. p(x) = xln xDerivatives Evaluate the derivatives of the following functions. 39. G(y) = ysin yDerivatives Evaluate the derivatives of the following functions. 40. Q(t) = t1/tMiscellaneous derivatives Compute the following derivatives using the method of your choice. 51. ddx(e10x2)Miscellaneous derivatives Compute the following derivatives using the method of your choice. 54. ddx(xe+ex)Miscellaneous derivatives Compute the following derivatives using the method of your choice. 50. ddx(x2x)24EMiscellaneous derivatives Compute the following derivatives using the method of your choice. 53. ddx((1x)x)Miscellaneous derivatives Compute the following derivatives using the method of your choice. 56. ddx(x(x10))Miscellaneous derivatives Compute the following derivatives using the method of your choice. 55. ddx(1+4x)xMiscellaneous derivatives Compute the following derivatives using the method of your choice. 57. ddx(cos(x2sinx))Integrals with ln x Evaluate the following integrals. Include absolute values only when needed. 13. 032x1x+1dxIntegrals Evaluate the following integrals. Include absolute values only when needed. x24x3+7dxIntegrals with ln x Evaluate the following integrals. Include absolute values only when needed. 15. ee2dxxln3xIntegrals with ln x Evaluate the following integrals. Include absolute values only when needed. 16. 0/2sinx1+cosxdxIntegrals with ln x Evaluate the following integrals. Include absolute values only when needed. 17. e2x4+e2xdxIntegrals with ln x Evaluate the following integrals. Include absolute values only when needed. 18. dxxlnxln(lnx)Integrals with ln x Evaluate the following integrals. Include absolute values only when needed. 19. e2e3dxxlnxln2(lnx)Integrals with ln x Evaluate the following integrals. Include absolute values only when needed. 20. 01yln4(y2+1)y2+1dyIntegrals with ex Evaluate the following integrals. 21. 024xex2/2dxIntegrals with ex Evaluate the following integrals. 22. esinxsecxdxIntegrals with ex Evaluate the following integrals. 23. exxdxIntegrals with ex Evaluate the following integrals. 24. 22ez/2ez/2+1dzIntegrals with ex Evaluate the following integrals. 25. ex+exexexdxIntegrals with ex Evaluate the following integrals. 26. ln2ln3ex+exe2x2+e2xdxIntegrals with general bases Evaluate the following integrals. 27. 1110xdxIntegrals with general bases Evaluate the following integrals. 28. 0/24sinxcosxdxIntegrals with general bases Evaluate the following integrals. 29. 12(1+lnx)xxdxIntegrals with general bases Evaluate the following integrals. 30. 1/31/2101/pp2dpIntegrals with general bases Evaluate the following integrals. 31. x26x3+8dxIntegrals with general bases Evaluate the following integrals. 32. 4cotxsin2xdxIntegrals Evaluate the following integrals. Include absolute values only when needed. ssssMiscellaneous integrals Evaluate the following integrals. 58. 72xdxMiscellaneous integrals Evaluate the following integrals. 59. 32xdxMiscellaneous integrals Evaluate the following integrals. 60. 0555xdxMiscellaneous integrals Evaluate the following integrals. 61. x210x3dxMiscellaneous integrals Evaluate the following integrals. 62. 02sinxcosxdxMiscellaneous integrals Evaluate the following integrals. 63. 12e3lnxxdxMiscellaneous integrals Evaluate the following integrals. 64. sin(lnx)4xdxMiscellaneous integrals Evaluate the following integrals. 65. 1e2(lnx)5xdxMiscellaneous integrals Evaluate the following integrals. 66. ln2x+2lnx1xdxMiscellaneous integrals Evaluate the following integrals. 67. 0ln2e3xe3xe3x+e3xdxIntegrals Evaluate the following integrals. Include absolute values only when needed. e2x(e2x1)2dxIntegrals Evaluate the following integrals. Include absolute values only when needed. e5+xxdxMiscellaneous integrals Evaluate the following integrals. 68. 0116x42xdxCalculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with IHpitals Rule. limh0(1+2h)1/hCalculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with IHpitals Rule. limh0(1+3h)2/hCalculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with IHpitals Rule. limx02x1xCalculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with IHpitals Rule. limx0ln(1+x)x67E68E69E70E