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All Textbook Solutions for Calculus Volume 1

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify / such that the integral takes the form f(u)du. 260. x ( 4 x 2 +9 )2dx=18( 4x2 +9);u=4x2+9 In the following exercises, find the antiderivative using the indicated substitution. 261. ( x+1)4dx;u=x+1 In the following exercises, find the antiderivative using the indicated substitution. 262. ( x1)5dx;u=x1 In the following exercises, find the antiderivative using the indicated substitution. 263. ( 2x3)7dx;u=2x3 In the following exercises, find the antiderivative using the indicated substitution. 264. ( 3x2)11dx;u=3x2 In the following exercises, find the antiderivative using the indicated substitution. 265. x x2 +1dx;u=x2+1 In the following exercises, find the antiderivative using the indicated substitution. 266. x 1x2 dx;u=1x2 In the following exercises, find the antiderivative using the indicated substitution. 267. (x1)( x2 2x)3dx;u=x22x In the following exercises, find the antiderivative using the indicated substitution. 268. (x22x)( x3 3x2 )2dx;u=x3=3x2 In the following exercises, find the antiderivative using the indicated substitution. 269. cos3d;u=sin(Hint: cos2=1 sin2)In the following exercises, find the antiderivative using the indicated substitution. 270. sin3d;u=cos(Hint: sin2=1 cos2)In the following exercises, use a suitable change of variables to determine the indefinite integral. 271. x( 1x)99dx In the following exercises, use a suitable change of variables to determine the indefinite integral. 272. t( 1t2 )10dt In the following exercises, use a suitable change of variables to determine the indefinite integral. 273. ( 11x7)3dx In the following exercises, use a suitable change of variables to determine the indefinite integral. 274. ( 7x11)4dx In the following exercises, use a suitable change of variables to determine the indefinite integral. 275. cos3sind In the following exercises, use a suitable change of variables to determine the indefinite integral. 276. sin7cosd In the following exercises, use a suitable change of variables to determine the indefinite integral. 277. cos2(t)sin(t)dt In the following exercises, use a suitable change of variables to determine the indefinite integral. 278. sin2xcos3xdx(Hint:sin2x+cos2x=1)In the following exercises, use a suitable change of variables to determine the indefinite integral. 279. tsin(t2)cos(t2)dt In the following exercises, use a suitable change of variables to determine the indefinite integral. 280. t2cos2(t3)sin(t3)dt In the following exercises, use a suitable change of variables to determine the indefinite integral. 280. x2 ( x 3 3 )2dx In the following exercises, use a suitable change of variables to determine the indefinite integral. 282. x3 1x2 dx In the following exercises, use a suitable change of variables to determine the indefinite integral. 283. y5 ( 1 y 3 ) 3/2 dy In the following exercises, use a suitable change of variables to determine the indefinite integral. 284. cos( 1cos)99sind In the following exercises, use a suitable change of variables to determine the indefinite integral. 285. ( 1 cos3 )10cos2sind In the following exercises, use a suitable change of variables to determine the indefinite integral. 286. (cos1)( cos2 2cos)3sind In the following exercises, use a suitable change of variables to determine the indefinite integral. 287. ( sin22sin)( sin3 3 sin2 )3cosd In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. 288. [T] y=3(1x)2 over [0, 2]In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. 289. [T] y=x(1x2)3 over [1,2]In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. 290. [T] y=sinx(1cosx)2over[0,]In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. 291. [T] y=x( x 2+1)2over[1,1]In the following exercises, use a change of variables to evaluate the definite integral. 292. 01x1x2dx In the following exercises, use a change of variables to evaluate the definite integral. 293. 01x 1+x2 dx In the following exercises, use a change of variables to evaluate the definite integral. 294. 02t 5+t2 dt In the following exercises, use a change of variables to evaluate the definite integral. 295. 01t2 1+t3 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] (2x+1)ex2+x6dxover[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 ))xdxover[0,2]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. 300. [T] 3x2+2x+1 x3 +x2 +x+4dxover[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 cos3xdxover[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+3dxover[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+1dxover[0,1]If h(a)=h(b) in abg(h(x))h(xdx) , what can you say about the value of the integral?Is the substitution u=1x2 in the definite integral 02x1x2dx okay? If not,why not 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. 0tcos(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 average 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 c0 .Find the area under the graph of f(t)=t(1+ t 2)a between t=0 and t=x where a0 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 0x1 and a0 is fixed. Evaluate the limit as x1 .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=1 to a and with a minor axis that is the y-axis from y=b to b can be written as aab1 x2 a2 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 .[T] The following graph is of a function of the form f(t)=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. etamxsec2xdx 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 tine integral. 342. Inxdx(Hint: Inxdx=12 xIn( x 2 )dx)In the following exercises, verify by differentiation that Inxdx=x(Inx1)+C , then use appropriate changes of variables to compute tine integral. 343. x2In2xdx In the following exercises, verify by differentiation that Inxdx=x(Inx1)+C , then use appropriate changes of variables to compute tine integral. 344. Inxx2dx(Hint:Setu=1x)In the following exercises, verify by differentiation that Inxdx=x(Inx1)+C , then use appropriate changes of variables to compute tine integral. 345. Inxxdx(Hint:Setu=x)Write an integral to express the area under the graph of y=1t from t=1 to ex and evaluate die 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( x2 )cos( x2 )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-end point approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate and solve for the exact area. 366. [T] y=exover[0,1]In the following exercises, does the right-end point approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate and solve for the exact area. 367. [T] y=exover[0,1]In the following exercises, does the right-end point approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate and solve for the exact area. 368. [T] y=In(x)over[1,2]In the following exercises, does the right-end point approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate and solve for the exact area. 369. [T] y=x+1x2+2x+6over[0,1]In the following exercises, does the right-end point approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate and solve for the exact area. 370. [T] y=2xover[1,0]In the following exercises, does the right-end point approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate and solve for the exact area. 371. [T] y=2xover[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 infinity, of the area under the graph of f(x)=xex2 between x=0 and x=5 .Show that abdtt= 1/b 1/a dt t 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 title to find the derivative of fg.Use the previous exercise to find the derivative of h(x)=xx(1+Inx) and evaluate 23xx(1+Inx)dx .Show that if c0 , then the integral of 1/x from ac to bc (0ab) is the same as the integral of 1/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 txy1tdt to show that Inxy=Inx+Iny for x, y > 0.Use the identity Inx=1xdtx show that In(x) is an increasing function of x on [0,) , and use the previous exercises to show that die range of In(x) is (,) . Without any further assumptions, conclude that In(x) has an inverse function defined on (,) .Pretend, for the moment, that we do not know that ex is the inverse function of In(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, for the moment, that we do not know that ex is die inverse function of Inx , 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 , 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 ( x+1 )2/8 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 391. 03/2dx 1x2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 392. 1/21/2dx 1x2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 393. 31dx 1+x2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 394. 1/33dx1+x2 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 395. 12dx|x| x2 1 In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 396. 12/3dx|x| x2 1 In the following exercises, find each indefinite integral, using appropriate substitutions. 397. dx 9x2 In the following exercises, find each indefinite integral, using appropriate substitutions. 398. dx 116x2 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| x2 9 In the following exercises, find each indefinite integral, using appropriate substitutions. 402. dx|x| 4x2 16 Explain the relationship cos1t+C=dt 1t2 =sin1t+C . Is it true, in general, that cos1t=sin1t?Explain the relationship sec1t+C=dt|t| t2 1=csc1t+C . Is it true, in general, that sec1t=csc1t Explain what is wrong with the following integral: 12dt 1t2 Explain what is wrong with the following integral: 11dt|t| t2 1 In the following exercises, solve for the antiderivative f of f with C = 0, then use a calculator to graph f and the antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative recovers the definite integral F(x)=axf(t)dt . 407. [T] 1 9x2 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 antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative recovers the definite integral F(x)=axf(t)dt . 408. [T] 9 9+x2 dx 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 antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative 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 antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative 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 1 1t2 dx In the following exercises, compute the antiderivative using appropriate substitutions. 412. dt sin 1t 1t2 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( t2 )1+t4dt In the following exercises, compute the antiderivative using appropriate substitutions. 415. sec 1( t2 )|t| t2 4dt In the following exercises, compute the antiderivative using appropriate substitutions. 416. t sec 1( t2 )t2 t4 1dt In the following exercises, use a calculator to graph the antiderivative f with C = 0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral F(x)=axf(t)dt . 471. 1x x2 4dx over [2, 6]In the following exercises, use a calculator to graph the antiderivative f with C = 0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral F(x)=axf(t)dt . 418. [T] 1( 2x+2)xdx over [0, 6]In the following exercises, use a calculator to graph the antiderivative f with C = 0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative 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, use a calculator to graph the antiderivative f with C = 0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral F(x)=axf(t)dt . 420. [T] 2e 2x1e4xdx over [0, 2]In the following exercises, use a calculator to graph the antiderivative f with C = 0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral F(x)=axf(t)dt . 421. [T] 1x+x In2xdx over [0, 2]In the following exercises, use a calculator to graph the antiderivative f with C = 0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral F(x)=axf(t)dt . 422. sin 1x 1x2 over [-1, 1]In the following exercises, compute each integral using appropriate substitutions. 423. ex 1e 2t dt In the following exercises, compute each integral using appropriate substitutions. 424. et1+e 2tdt In the following exercises, compute each integral using appropriate substitutions. 425. dtt 1 In2 t In the following exercises, compute each integral using appropriate substitutions. 426. dtt( 1+ In2 t)In the following exercises, compute each integral using appropriate substitutions. 427. cos 1( 2t) 14t2 dt In the following exercises, compute each integral using appropriate substitutions. 428. et cos 1( et ) 1e 2t dt In the following exercises, compute each definite integral. 429. 01/2tan( sin 1 t) 1t2 dt In the following exercises, compute each definite integral. 430. 1/41/2tan( cos 1 t) 1t2 dt In the following exercises, compute each definite integral. 431. 01/2sin( tan 1 t)1+t2dt In the following exercises, compute each definite integral. 432. 01/2cos( tan 1 t)1+t2dt For A0 , compute I(A)=AAdt1+t2 and evaluate limaI(A) , the area under the graph of 11+t2 on [,] .For 1B , compute I(B)=1Bdtt t2 1 and evaluate limBI(B) , the area under the graph of 1tt21 over [1,) .Use the substitution u=2cotx and the identity 1+cot2x=csc2x to evaluate dx1+ cos2x . (Hint: Multiply the top and the bottom of the integrand by csc2x .)[T] 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 over their domains. 439. If f(x)0,f(x)0 . f all x, then the righthand 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 over their domains. 440. a 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 over their domains. 441. If f(x)g(x) for all x[a,b] , then abf(x)abg( x) .All continuous functions have 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.y=3x22x+1 over [1,1]y=In(x2+1) over [0,e]y=x2sinx over [0,]y=x+1x over [1,4]Evaluate the following integrals. 447. 11(x32x2+4x)dx Evaluate the following integrals. 448. 043t 1+6t2 dt Evaluate the following integrals. 449. /3/22sec(2)tan(2)d Evaluate the following integrals. 450. 0/4ecos 2 xsinxcosdxy ^r/4 2 £'C°!' Tsinxcosdr Find the antiderivative. 451. dx ( x+4 )3 Find the antiderivative. 452. xIn(x2)dx Find the antiderivative. 453. 4x2 1x6 dx Find the antiderivative. 454. e 2x1+e 4xdx Find the derivative. 455. ddt0tsinx 1+x2 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 -5,468,750 1990 -755,495 1995 -73,005 2000 -29768 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 -5,468,750 1990 -755,495 1995 -73,005 2000 -29768 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$. Find the average cost per gigabyte of RAM for 1980 to 2010.The following problems consider the historic average cost per gigabyte of RAM on a computer. Year 5-Year Change ($) 1980 0 1985 -5,468,750 1990 -755,495 1995 -73,005 2000 -29768 2005 -918 2010 -177 461. Find die average cost of 1GB RAM for 2005 to 2010.The velocity of a bullet from a rifle 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?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. 2. 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=x3andy=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=cosandy=0.5,for0 For the following exercises, determine the area of the region between the two curves by integrating over the y-axis. 5. x=y2andx=9 For the following exercises, determine the area of the region between the two curves by integrating over the y-axis. 6. y=xandx=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=exx=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|andy=x2 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. 14. y=sin(x),y=2x,andx0 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. 15. y=12x,y=x,andy=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. 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. Determine its area by integrating over the x-axis. 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. Determine its area by integrating over the x-axis. 18. y=x2+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. Determine its area by integrating over the x-axis. 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 , 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=ex 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=y3 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. 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] x=4y2 and x=1+3y+y2 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 die 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)=(1/2)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 is given by the sinusoidal function H(t)=(1/2)(1/2)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 yon 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.wwFor 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.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=1x2 and 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 legion 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,x=1, 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, and y=1 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,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. 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=3x2,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=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. 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=0 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 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=

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