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All Textbook Solutions for Calculus Volume 2

In the following exercises, use a change of variables to evaluate the definite integral. 294. 02t 5 t 2 dt In the following exercises, use a change of variables to evaluate the definite integral. 295. 01t 1+ t 3 dt In the following exercises, use a change of variables to evaluate the definite integral. 296. 0/4sec2tand In the following exercises, use a change of variables to evaluate the definite integral. 297. 0/4sin cos4d In the following exercises, evaluate the indefinite integral f(x)dx with constant C = 0 using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral F(x)=axf(t)dt , with a the left endpoint of the given interval. 298. [T] f(2x+1)ex2+x6dx over [3, 2]In the following exercises, evaluate the indefinite integral f(x)dx with constant C = 0 using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral F(x)=axf(t)dt , with a the left endpoint of the given interval. 299. [T] cos( In( 2x ))xdx on [0, 21]In the following exercises, evaluate the indefinite integral f(x)dx with constant C = 0 using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral F(x)=axf(t)dt , with a the left endpoint of the given interval. 300. [T] 3x2+2x+1 x 3 + x 2 +x+4dx over [1, 2]In the following exercises, evaluate the indefinite integral f(x)dx with constant C = 0 using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral F(x)=axf(t)dt , with a the left endpoint of the given interval. 301. [T] sinx cos3xdx over [3,3]In the following exercises, evaluate the indefinite integral f(x)dx with constant C = 0 using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral F(x)=axf(t)dt , with a the left endpoint of the given interval. 302. [T] (x+2)ex24x+3dx over [5, 1]In the following exercises, evaluate the indefinite integral f(x)dx with constant C = 0 using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral F(x)=axf(t)dt , with a the left endpoint of the given interval. 303. [T] 3x22x3+1dx over [0, 1]If h(a)=h(b) in abg(h(x))h(x)dx , what can you say about the value Elf the integral?Is the substitution u=1x2 02x1x2dx okay? If not, why not in the definite integral In the following exercises, use a change of variables to show that each definite integral is equal to zero. 306. 0cos2(2)sin(2)d In the following exercises, use a change of variables to show that each definite integral is equal to zero. 307. 0tcot(t2)sin(t2)dt In the following exercises, use a change of variables to show that each definite integral is equal to zero. 308. 01(12t)dt In the following exercises, use a change of variables to show that each definite integral is equal to zero. 309. 0112t( 1+ ( t 1 2 ) 2 )dt In the following exercises, use a change of variables to show that each definite integral is equal to zero. 310. 0sin( ( t 2 )3)cos(t2)dt In the following exercises, use a change of variables to show that each definite integral is equal to zero. 311. 02(1t)cos(t)dt In the following exercises, use a change of variables to show that each definite integral is equal to zero. 312. /43/4sin2tcostdt Show that the avenge value of f(x) over an interval [a, b] is the same as the average value of f(cx) over the interval [ac,bc] for C > 0.€314. Find the area under the graph of f(t)=t(1 t 2)a between t = 0 and t = x where a > 0 and a1 is fixed, and evaluate the limit as x .Find the area under the graph of g(t)=t(1 t 2)a between t = 0 and t = x, where 0 < x < 1 and a > 0 is fixed. Evaluate the limit as x l.The area of a semicircle of radius 1 can be expressed as 111x2dx . Use the substitution x = cost to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral.The area of the top half of an ellipse with a major axis that is the x-axis from x = l to a and with a minor axis that is the y—axis from y = b to b can be written as aab1 x 2 a 2 dx . Use the substitution x = acost to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.[T] The following graph is of a function of the form f(t)=asin(nt)+bsin(mt) . Estimate the coefficients a and b, and the frequency parameters n and m. Use these estimates to approximate 0f(t)dt .The following graph is of a function of the form f(x)=acos(nt)+bcos(mt). Estimate the coefficients a and b and the frequency parameters n and m. Use these estimates to approximate 0f(t)dt.In the following exercises, compute each indefinite integral. 320. e2xdx In the following exercises, compute each indefinite integral. 321. e3xdx In the following exercises, compute each indefinite integral. 322. 2xdx In the following exercises, compute each indefinite integral. 323. 3xdx In the following exercises, compute each indefinite integral. 324. 12xdx In the following exercises, compute each indefinite integral. 325. 2xdx In the following exercises, compute each indefinite integral. 326. 1x2dx In the following exercises, compute each indefinite integral. 327. 1xdx In the following exercises, find each indefinite integral by using appropriate substitutions. 328. Inxxdx In the following exercises, find each indefinite integral by using appropriate substitutions. 329. dxx ( Inx )2 In the following exercises, find each indefinite integral by using appropriate substitutions. 330. dxxInx(x1)In the following exercises, find each indefinite integral by using appropriate substitutions. 331. dxxInxIn( Inx)In the following exercises, find each indefinite integral by using appropriate substitutions. 332. tand In the following exercises, find each indefinite integral by using appropriate substitutions. 333. cosxxsinxxcosxdx In the following exercises, find each indefinite integral by using appropriate substitutions. 334. In( sinx)tanxdx In the following exercises, find each indefinite integral by using appropriate substitutions. 335. In(cosx)tanxdx In the following exercises, find each indefinite integral by using appropriate substitutions. 336. xex2dx In the following exercises, find each indefinite integral by using appropriate substitutions. 337. x2ex3dx In the following exercises, find each indefinite integral by using appropriate substitutions. 338. esinxcosxdx In the following exercises, find each indefinite integral by using appropriate substitutions. 339. etanxsec2xdx In the following exercises, find each indefinite integral by using appropriate substitutions. 340. eInxdxx In the following exercises, find each indefinite integral by using appropriate substitutions. 341. e In( 1t )1tdt In the following exercises, verify by differentiation that Inxdx=x(Inx1)+C , then use appropriate changes of variables to compute the integral. 342. Inxdx (Hint: Inxdx=12xIn(x2)dx )In the following exercises, verify by differentiation that Inxdx=x(Inx1)+C , then use appropriate changes of variables to compute the integral. 343. x2In2xdx In the following exercises, verify by differentiation that Inxdx=x(Inx1)+C , then use appropriate changes of variables to compute the integral. 344. Inxx2dx (Hint: Set u=1x .)In the following exercises, verify by differentiation that Inxdx=x(Inx1)+C , then use appropriate changes of variables to compute the integral. 345. Inxxdx (Hint: Set u=x .)Write an integral to express the area under the graph of y=1t from t = l to ex and evaluate the integral.Write an integral to express the area under the graph of y = et between t = 0 and t = Inx, and evaluate the integral.In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 348. tan(2x)dx In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 349. sin( 3x)cos( 3x)sin( 3x)+cos( 3x)dx In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 350. xsin( x 2 )cos( x 2 )dx In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 351. xcsc(x2)dx In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 352. In(cosx)tanxdx In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 353. In(cscx)cotxdx In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 354. exe xex+e xdx In the following exercises, evaluate the definite integral. 355. 121+2x+x23x+3x2+x3dx In the following exercises, evaluate the definite integral. 356. 0/4tanxdx In the following exercises, evaluate the definite integral. 357. 0/3sinxcosxsinx+cosxdx In the following exercises, evaluate the definite integral. 358. /6/2cscxdx In the following exercises, evaluate the definite integral. 359. /4/3cotxdx In the following exercises, integrate using the indicated substitution. 360. xx100dx;u=x100 In the following exercises, integrate using the indicated substitution. 361. y1y+1dy;u=y+1 In the following exercises, integrate using the indicated substitution. 362. 1x23xx3dx;u=3xx3 In the following exercises, integrate using the indicated substitution. 363. sinx+cosxsinxcosxdx;u=sinxcosx In the following exercises, integrate using the indicated substitution. 364. e2x1e 2xdx;u=e2x In the following exercises, integrate using the indicated substitution. 365. In(x) 1 ( Inx ) 2 xdx;u=Inx In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. 366. [T] y=ex over [0, l]In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. 367. [T] y=ex over [0, 1]In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. 368. [T] y=In(x) over [1, 2]In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. 369. [T] y=x+1x2+2x+6 over [0, 1]In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. 370. [T] y=2x over [1, 0]In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. 371. [T] y=2x over [0, 1]In the following exercises, f(x)0 for axb . Find the area under the graph of f (x) between the given values a and b by integrating. 372. f(x)=log10(x)x;a=10,b=100 In the following exercises, f(x)0 for axb . Find the area under the graph of f (x) between the given values a and b by integrating. 373. f(x)=log2(x)x;a=32,b=64 In the following exercises, f(x)0 for axb . Find the area under the graph of f (x) between the given values a and b by integrating. 374. f(x)=2x;a=1,b=2 In the following exercises, f(x)0 for axb . Find the area under the graph of f (x) between the given values a and b by integrating. 375. f(x)=2x;a=3,b=4 Find the area under the graph of the function f(x)=xex2 between x = 0 and x = 5.Compute the integral of f(x)=xex2 and find the smallest value of N such that the area under the graph f(x)=xex2 between x = N and x=N+10 is at most, 0.01.Find the limit, as N tends to in?nity, of the area under the graph of f(x)=xex2 between x = 0 and x = 5.Show that abdtt=1/b1/adtt when 0ab .Suppose that f(x) > 0 for all x and that f and g are differentiable. Use the identity fg=egInf and the chain rule to find the derivative of fg.Use the previous exercise to find the antiderivative of h(x)=xx(1+Inx) and evaluate 23xx(1+Inx)dx .Show that if c > 0, then the integral of l/x from ac to bc (0 < a < b) is the same as the integral of l/x from a to b.The following exercises are intended to derive the fundamental properties of the natural log starting from the definition In(x)=1xdtt , using properties of the definite integral and making no further assumptions. 383. Use the identity In(x)=1xdtt to derive the identity In(1x)=Inx .The following exercises are intended to derive the fundamental properties of the natural log starting from the definition In(x)=1xdtt , using properties of the definite integral and making no further assumptions. 384. Use a change of variable in the integral 1xy1tdt to show that Inxy=Inx+Iny for x, y > 0.The following exercises are intended to derive the fundamental properties of the natural log starting from the definition In(x)=1xdtt , using properties of the definite integral and making no further assumptions. 385. Use the identity Inx=1xdtx to show that In(x) is an increasing function of x on [0,) , and use the previous exercises to show that the range of In(x) is (,) . Without any further assumptions, conclude that In(x) has an inverse function defined on (,) .Pretend, fat the moment, that we do not know that ex is the inverse function of ln(x), but keep in mind that In(x) has an inverse function defined on (,) . Call it E. Use the identity Inxy=Inx+Iny to deduce that E(a+b)=E(a)E(b) for any real numbers a, b.Pretend, fur the moment, that we do not know that ex is the inverse function of lnx, but keep in mind that Inx has an inverse function defined on (,) . Call it E. Show that E(t)=E(t) .The sine integral, defined as S(x)=0xsinttdt is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large x. Show that for k1,|S(2k)S(2(k+1))|1k(2k+1) . (Hint: sin(t+)=sint )[T] The normal distribution in probability is given by p(x)=12e(x)2/22 where is the standard deviation and is the average. The standard normal distribution in probability, ps, corresponds to =0 and =1 . Compute the left endpoint estimates R10 and R100 of 111 2ex 2/2dx .[T] Compute the right endpoint estimates R50 and R100 of 3512 2e ( x1 )2/8 .In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 391. 03/2dx 1 x 2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 392. 1/21/2dx 1 x 2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 393. 31dx 1 x 2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 394. 133dx1+x2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 395. 12dx|x| x 2 1 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 396. 12/3dx|x| x 2 1 In the following Exercises, find each indefinite integral, using appropriate substitutions. 397. dx 9 x 2 In the following Exercises, find each indefinite integral, using appropriate substitutions. 398. dx 116 x 2 In the following Exercises, find each indefinite integral, using appropriate substitutions. 399. dx9+x2 In the following Exercises, find each indefinite integral, using appropriate substitutions. 400. dx25+16x2 In the following Exercises, find each indefinite integral, using appropriate substitutions. 401. dx|x| x 2 9 In the following Exercises, find each indefinite integral, using appropriate substitutions. 402. dx|x| 4 x 2 16 Explain the relationship cos1t+C=dt 1 t 2 =sin1t+C . Is it true, in general, that cos1t=sin1t ?Explain the relationship sec1+C=dt|t| t 2 1=csc1t+C . Is it true, in general, that sec1t=csc1t ?Explain what is wrong with the following integral: 12dt 1 t 2 Explain what is wrong with the following integral: 11dt|t| t 2 1 .In the following exercises, solve for the antiderivative f of f with C = 0, then use a calculator to graph f and the antidefivative ever the given interval [a, b]. Identify a value of C such that adding C to the antidelivative recovers the definite integral F(x)=axf(t)dt . 407. [T] 1 9 x 2 dx over [—3, 3]In the following exercises, solve for the antiderivative f of f with C = 0, then use a calculator to graph f and the antidefivative ever the given interval [a, b]. Identify a value of C such that adding C to the antidelivative recovers the definite integral F(x)=axf(t)dt . 408. [T] 99+x2dx over [6, 6]In the following exercises, solve for the antiderivative f of f with C = 0, then use a calculator to graph f and the antidefivative ever the given interval [a, b]. Identify a value of C such that adding C to the antidelivative recovers the definite integral F(x)=axf(t)dt . 409. [T] cosx4+ sin2xdx over [6, 6]In the following exercises, solve for the antiderivative f of f with C = 0, then use a calculator to graph f and the antidefivative ever the given interval [a, b]. Identify a value of C such that adding C to the antidelivative recovers the definite integral F(x)=axf(t)dt . 410. [T] ex1+e 2xdx over [6, 6]In the following Exercises, compute the antiderivative using appropriate substitutions. 411. sin 1tdt 1 t 2 In the following Exercises, compute the antiderivative using appropriate substitutions. 412. dt sin 1t 1 t 2 In the following Exercises, compute the antiderivative using appropriate substitutions. 413. tan 1( 2t)1+4t2dt In the following Exercises, compute the antiderivative using appropriate substitutions. 414. t tan 1( t 2 )1+t4dt In the following Exercises, compute the antiderivative using appropriate substitutions. 415. sec 1( t 2 )|t| t 2 4dt In the following Exercises, compute the antiderivative using appropriate substitutions. 416. t sec 1( t 2 )t2 t 4 1dt In the following exercises, solve for the antiderivative f of with C = 0, the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antidelivative gives the same value as the definite integral F(x)=axf(t)dt . 417. [T] 1x x 2 4dx over [2, 6]In the following exercises, solve for the antiderivative f of with C = 0, the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antidelivative gives the same value as the definite integral F(x)=axf(t)dt . 418. [T] 1( 2x+2)xdx over [2, 6]In the following exercises, solve for the antiderivative f of with C = 0, the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antidelivative gives the same value as the definite integral F(x)=axf(t)dt . 419. [T] ( sinx+xcosx)1+x2 sin2xdx over [6, 6]In the following exercises, solve for the antiderivative f of with C = 0, the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antidelivative gives the same value as the definite integral F(x)=axf(t)dt . 420. [T] 2e 2x 1 e 4x dx over [0, 2]In the following exercises, solve for the antiderivative f of with C = 0, the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antidelivative gives the same value as the definite integral F(x)=axf(t)dt . 421. [T] 1x+x In2x over [0, 2]In the following exercises, solve for the antiderivative f of with C = 0, the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antidelivative gives the same value as the definite integral F(x)=axf(t)dt . 422. [T] sin 1x 1 x 2 over [1, 1]In the following exercises, compute each integral using appropriate substitutions. 423. ex 1 e 2t dt In the following exercises, compute each integral using appropriate substitutions. 424. et 1+ e 2t dt In the following exercises, compute each integral using appropriate substitutions. 425. dtt 1 In 2 t In the following exercises, compute each integral using appropriate substitutions. 426. dtt( 1+ In 2 t)In the following exercises, compute each integral using appropriate substitutions. 427. cos 1( 2t) 14 t 2 dt In the following exercises, compute each integral using appropriate substitutions. 428. et cos 1( e t ) 1 e 2t dt In the following Exercises, compute each definite integral. 429. 01/2tan( sin 1 t) 1 t 2 dt In the following Exercises, compute each definite integral. 430. 1/41/2tan( cos 1 t) 1 t 2 dt In the following Exercises, compute each definite integral. 431. 01/2sin( tan 1 t)1t2dt In the following Exercises, compute each definite integral. 432. 01/2cos( tan 1 t)1t2dt For A > 0, compute I(A)=AAdt1+t2 and evaluate limaI(A) , the area under the graph of 11+t2 on [,] .For 1B , compute I(B)=1Bdtt t 2 1 and evaluate limBI(B) , the area under the graph of 1tt21 on [1,) .Use the substitution u=2cotx and the identity 1+cot2x=csc2x to evaluate dx1+ cos2x . (Hint: Multiply the top and bottom of the integrand by csc2 x.)Approximate the points at which the graphs of f(x)=2x21 and g(x)=(1+4x2)3/2 intersect, and approximate the area between their graphs accurate to three decimal places.. [T] Approximate the points at which the graphs of f(x)=x21 and f(x)=x21 intersect, and approximate the area between their graphs accurate to three decimal places.Use the following graph to prove that 0x1t2dt=12x1x2+12sin1x .True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous aver their domains. 439. If f(x)0,f(x)0 for all x, then the right-hand rule underestimates the integral abf(x) . Use a graph to justify your answer.True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous aver their domains. 440. abf(x)2dx=abf(x)dxabf(x)dx True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous aver their domains. 441. If f(x)g(x) for all x[a,b] , then abf(x)abg(x) .True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous aver their domains. 442. All continuous functions have an antiderivative.Evaluate the Riemann sums L4 and R4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. 443. y=3x22x+1 over [1, 1]Evaluate the Riemann sums L4 and R4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. 444. y=In(x2+1) over [0, e]Evaluate the Riemann sums L4 and R4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. 445. y=x2sinx over [0, ]Evaluate the Riemann sums L4 and R4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. 446. y=x+1x over [1, 4]Evaluate the following integrals. 447. 11(x32x2+4x)dx Evaluate the following integrals. 448. 043t 1+6 t 2 dt Evaluate the following integrals. 449. /3/22sec(2)tan(2)d Evaluate the following integrals. 450. 0/4e cos2xsinxcosdx Find the antiderivative. 451. dx ( x+4 )3 Find the antiderivative. 452. xIn(x2)dx Find the antiderivative. 453. 4x2 1 x 6 dx Find the antiderivative. 454. e 2x1+e 4xdx Find the derivative. 455. ddt0tsinx 1+ x 2 dx Find the derivative. 456. ddx1x34t2dt Find the derivative. 457. ddx1In(x)(4t+et)dt Find the derivative. 458. ddx0cosxet2dt The following problems consider the historic average cost per gigabyte of RAM on a computer. Year 5-Year Change ($) 1980 0 1985 5,458,750 1990 755,495 1995 73,005 2000 29,768 2005 918 2010 177 459. If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.The following problems consider the historic average cost per gigabyte of RAM on a computer. Year 5-Year Change ($) 1980 0 1985 5,458,750 1990 755,495 1995 73,005 2000 29,768 2005 918 2010 177 460. The average cost per gigabyte of RAM can be approximated by the function C(t)=8,500,000(0.65)t , where t is measured in years since 1980, and C is cost in US$. Find the average cost per gigabyte of RAM for 1930 to 2010.The following problems consider the historic average cost per gigabyte of RAM on a computer. Year 5-Year Change ($) 1980 0 1985 5,458,750 1990 755,495 1995 73,005 2000 29,768 2005 918 2010 177 461. Find the average cost of 1GB RAM for 2005 to 2010.The following problems consider the historic average cost per gigabyte of RAM on a computer. Year 5-Year Change ($) 1980 0 1985 5,458,750 1990 755,495 1995 73,005 2000 29,768 2005 918 2010 177 462. The velocity of a bullet from a rifie can be approximated by v(t)=6400t26505t+2686 , where t is seconds after the shot and v is the velocity measured in feet per second. This equation only models the velocity for the first half—second after the shot: 0t0.5 . What is the total distance the bullet travels in 0.5 sec?The following problems consider the historic average cost per gigabyte of RAM on a computer. Year 5-Year Change ($) 1980 0 1985 5,458,750 1990 755,495 1995 73,005 2000 29,768 2005 918 2010 177 463. What is the average velocity of the bullet for the first half-second?For the following exercises, determine the area of the region between the two curves in the given figure by integrating over the x-axis. y=x23 and y=1 For the following exercises, determine the area of the region between the two curves in the given figure by integrating over the x-axis. y=x2 and y=3x+4 For the following exercises, split the region between the two curves into two smaller regions, then determine the area by integrating over the x-axis. Note that you will have two integrals to solve. 3. y=x3 and y=x2+x For the following exercises, split the region between the two curves into two smaller regions, then determine the area by integrating over the x-axis. Note that you will have two integrals to solve. 4. y=cos and y=0.5 , for 0 For the following exercises, determine the area of the region between the two curves by integrating over the y-axis. 5. x=y2 and x=9 For the following exercises, determine the area of the region between the two curves by integrating over the y-axis. 6. y=x and x=y2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 7. y=x2 and y=x2+18x For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 8. y=1x,y=1x2 , and x=3 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 9. y=cosx and y=cos2x on x=[,]For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 10. y=ex,y=e2x1 , and x=0 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 11. y=ex,y=ex,x=1 and x=1 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 12. y=e,y=ex, and y=ex For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 13. y=x and y=x2 For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. 14. y=sin(x),y=2x and x0 For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. 15. y=12x,y=x , and y=1 For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. 16. y=sinx and y=cosx over x=[,]For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. 17. y=x3 and y=x22x over x=[1,1]For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. 18. y=x3+9 and y=10+2x over x=[1,3]For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area. 19. y=x3+3x and y=4x For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y-axis. 20. x=y3 and x=3y2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y-axis. 21. x=2y and x=y3y For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y-axis. 22. x=3+y2 and x=yy2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y-axis. 23. y2=x and x=y+2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y-axis. 24. x=y and 2x=y2+2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y-axis. 25. x=siny,x=cos(2y),y=/2 and y=/2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 26. x=y4 and x=y5 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 27. y=xex,y=ex,x=0 and x=1 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 28. y=x6 and y=x4 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 29. x=y3+2y2+1 and x=y2+1 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 30. y=x and y=x21 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 31. y=43x and y=1x For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 32. y=sinx,x=/6 and y=cos3x For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 33. y=x23x+2 and y=x32x2x+2 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 34. y=2cos3(3x),y=1,x=4 and x=4 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 35. y=y3=x and 2y=x For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 36. y=1x2 and y=x21 For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. 37. y=cos1x,y=sin1x,x=1 and x=1 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 38. [T] x=ey and y=x2 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 39. [T] y=x2 and y=1x2 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 40. [T] y=3x2+8x+9 and 3y=x+24 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 41. [T] x=4y2 and y2=1+x2 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 42. [T] x2=y2 and x=3y For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 43. [T] y=sin3x+2,y=tanx,x=1.5 , and x=1.5 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 44. [T] y=1x2 and y2=x2 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 45. [T] y=1x2 and y=x2+2x+1 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 46. [T] y=4y2 For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 47. [T] y=cosx,y=ex , x= and x=0 The largest triangle with a base on the x- axis that fits inside the upper half of the unit circle y2+x2=1 is given by y=1+x and y=1x. See the following figure. What is the area inside the semicircle but outside the triangle?A factory selling cell phones has a marginal cost function C(x)=0.01x23x+229 , where x represents the number of cell phones, and a marginal revenue function given by R(x)=4292x. Find the area between the graphs of these curves and x=0. What does this area represent?An amusement park has a marginal cost function C(x)=1000ex+5 , where x represents the number of tickets sold, and a marginal revenue function given by R(x)=600.1x. Find the total profit generated when Selling 550 tickets. Use a calculator to determine intersection points, if necessary, to two decimal places.The tortoise versus the hare: The speed of the hare is given by the sinusoidal function H(t)=1cos((t)/2) whereas the speed of the tortoise is T(t)=(l/2)tan1(t/4) ,where t is time measured in hours and the speed is measured in miles per hour. Find the area between the curves from time t=0 to the first time after one hour when the tortoise and hare are traveling at the same speed. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.The tortoise versus the hare: The speed of the hare isgivenbythesinusoidalfunction H(t)=(1/2)(l/2)cos(2t) whereas the speed of the tortoise is T(t)=t , where t is time measured in hours and speed is measured in kilometers per hour. If the race is over in 1 hour, who won the race and by how much? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.For the following exercises, find the area between the curves by integrating with respect to x and then with respect to y. Is one method easier than the other? Do you obtain the same answer? 53. y=x2+2x+1 and y=x23x+4 For the following exercises, find the area between the curves by integrating with respect to x and then with respect to y. Is one method easier than the other? Do you obtain the same answer? 54. y=x4 and x=y5 For the following exercises, find the area between the curves by integrating with respect to x and then with respect to y. Is one method easier than the other? Do you obtain the same answer? 55. x=y22 and x=2y For the following exercises, solve using calculus, then check your answer with geometry. 56. Determine the equations for the sides of the square that touches the unit circle on all four sides, as seen in the following figure. Find the area between the perimeter of this square and the unit circle. Is there another way to solve this without using calculus?For the following exercises, solve using calculus, then check your answer with geometry. 57. Find the area between the perimeter of the unit circle and the triangle created from y=2x+1 , y=12x and y=35 , as seen in the following figure. Is there a way to solve this without using calculus?Derive the formula for the volume of a sphere using the slicing method.Use the slicing method to derive the formula for the volume of a cone.Use the slicing method to derive the formula for the volume of a tetrahedron with side length a.Use the disk method to derive the formula for the volume of a trapezoidal cylinder.Explain when you would use the disk method versus the washer method. When are they interchangeable?For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume. 63. A pyramid with height 6 units and square base of side 2 units, as pictured here.For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume. 64. A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume. 65. A tetrahedron with a base side of 4 units, as seen here.For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume. 66. A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume. 67. A cone of radius r and height h has a smaller cone of radius r/2 and height h/2 removed from the top, as seen here. The resulting solid is called a frustum.For the following exercises, draw an outline of the solid and find the volume using the slicing method. 68. The base is a circle of radius a. The slices perpendicular to the base are squares.For the following exercises, draw an outline of the solid and find the volume using the slicing method. 69.The base is a triangle with vertices (0, 0), (1, 0), and (0, 1). Slices perpendicular to the xy-plane are semicircles.For the following exercises, draw an outline of the solid and find the volume using the slicing method. 70. The base is the region under the parabola y=1x2 in the first quadrant. Slices perpendicular to the xy-plane are squares.For the following exercises, draw an outline of the solid and find the volume using the slicing method. 71. The base is the region under the parabola y=1x2and above the x-axis. Slices perpendicular to the y-axis are squares.For the following exercises, draw an outline of the solid and find the volume using the slicing method. 72. The base is the region enclosed by y=x2 and y = 9. Slices perpendicular to the x-axis are right isosceles triangles.For the following exercises, draw an outline of the solid and find the volume using the slicing method. 73.The base is the area between y = x and y=x2. Slices perpendicular to the x-axis are semicircles.For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 74. x+y=8 , x=0 , and y=0 For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 75. y=2x2 , x=0 , x=4 , and y=0 For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 76. y=ex+1 , x=0 , and y=0 For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 77. y=x4 , x=0 , x=4 , and y=0 For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 78. y=x , x=0 , and y=0 For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 79. y=sinx,y=cosx and x=0 For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 80. y=1x,x=2 , and y=3 For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis. 81. x2y2=9 and x+y=9 , y=0 and x=0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 82. y=412x,x=0 and y=0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 83. y=2x3,x=0,x=1 and y=0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 84. y=3x3,x=0 , and y=3 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 85. y=4x2,y=0 , and x=0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 86. y=1x+1,x=0 , and x=0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 87. x=sec(y) and y=4,y=0 and x=0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 88. y=1x+1,x=0 , and x=2 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 89. y=4x,y=x and x=2 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 90. y=x+2 , y=x+6 , x=0 , and x=5 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 91. y=x2 and y=x+2 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 92. x2=y3+2 and x3=y2 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 93. y=4x2 and y=2x For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 94. [T] y=cosx,y=ex,x=0 , and x=1.2927 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 95. y=x and y=x2 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 96. y=sinx,y=5sinx,x=0 and x=For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 97. y=1+x2 and y=4x2 For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 98. y=x , x=4 , and y=0 For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 99. y=x+2 , y=2x1 , and x=0 For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 100. y=x3 and y=x3 For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 101. x=e2y,x=y2,y=0 , and y=ln(1)For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 102. x=9y2,x=ey,y=0 , and y=3 Yogurt containers can be shaped like frustums. Rotate the line y=1mxaround the y-axis to find the volume between y=a and y = b.Rotate the ellipse (x2/a2)+(y2/b2)=1 around the x-axis to approximate the volume of a football, as seen here.Rotate the ellipse (x2/a2)+(y2/b2)=1 around the y-axis to approximate the volume of a football.A better approximation of the volume of a football is given by the solid that comes from rotating y=sin x around the x-axis from x=0 to x=. What is the volume of this football approximation, as seen here?What is the volume of the Bundt cake that comes from rotating y = sin x around the y-axis from x = 0 to x= ?For the following exercises, find the volume of the solid described. 110. Bore a hole of radius a down the axis of a right cone and through the base of radius bfas seen here.For the following exercises, find the volume of the solid described. 109. The base is the region enclosed by the generic ellipse (x2/a2)+(y2/b2)=1 . Slices perpendicular to the x-axis are semicircles.For the following exercises, find the volume of the solid described. 110. Bore a hole of radius a down the axis of a right cone and through the base of radius b,as seen here For the following exercises, find the volume of the solid described. 111. Find the volume common to two spheres of radius r with centers that are 2h apart, as shown here.For the following exercises, find the volume of the solid described. 112. Find the volume of a spherical cap of height h and radius r where h < r, as seen here.For the following exercises, find the volume of the solid described. 113. Find the volume of a sphere of radius R with a cap of height h removed from the top, as seen here.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 114.[T] Over the curve of y=3x, x=0 ,and y = 3 rotated around the y-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 115.[T] Under the curve of y=3x, x=0 , and x = 3 rotated around the y-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 116. [T] Over the curve of y=3x , x=0 , and y = 3 rotated around the x-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 117.[T] Under the curve of y=3x, x=0 , and x = 3 rotated around the x-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 118.[T] Under the curve of y=2x3, x=0 , and x=2 rotated around the y-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 119.[T] Under the curve of y=2x3, x=0, and x = 2 rotated around the x-axis.For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 120. y=1x2 , x=0 , and x=1 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 121. y=5x3 , x=0 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 122. y=1x,x=1 and x=100 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 123. y=1x2,x=0 and x=3 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 124. y=11+x2,x=0 and x=3 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 125. y=sinx2,x=0 , and x=For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 126. y=11x2,x=0 , and x=12 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 127. y=x,x=0 , and x=1 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 128. y=(1+x2)3,x=0 , x=1 For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 129. y=5x32x4,x=0 and x=2 For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 130. y=1x2,x=0 , and x=1

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