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All Textbook Solutions for Single Variable Calculus: Early Transcendentals
27E28E29E30E31E32E(a) Newtons Law of Gravitation states that two bodies with masses m1, and m2, attract each other with a force F=Gm1m2r2 where r is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from r = a to r = b. (b) Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earths mass is 5.98 1024 kg and is concentrated at its center. Take the radius of the earth to be 6.37 106 m and G = 6.67 1011 Nm2/kg234EFind the average value of the function on the given interval. f(x) = 3x2 + 8x, [1, 2]2EFind the average value of the function on the given interval. g(x) = 3 cos x, [/2, /2]4EFind the average value of the function on the given interval. f(t) = esin t cos t, [0, /2]6EFind the average value of the function on the given interval. h(x) = cos4x sin x, [0, ]8E(a) Find the average value of f on the given interval. (b) Find c such that fave = f(c). (c) Sketch the graph off and a rectangle whose area is the same as the area under the graph of f. f(x) = (x 3)2, [2, 5]10E11E12EIf f is continuous and 13f(x)dx=8, show that f takes on the value 4 at least once on the interval [1, 3].14EFind the average value of f on [0, 8].The velocity graph of an accelerating car is shown. (a) Use the Midpoint Rule to estimate the average velocity of the car during the first 12 seconds. (b) At what time was the instantaneous velocity equal to the average velocity?In a certain city the temperature (in F) t hours after 9 am was modeled by the function T(t)=50+14sint12 Find the average temperature during the period from 9 am to 9 pm.18EThe linear density in a rod 8 m long is 12/x+1kg/m, where x is measured in meters from one end of the rod. Find the average density of the rod.20E21E22E23E24E25E26E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC1RQ2RQ3RQ4RQ5RQ6RQ7RQ8RQ9RQ10RQ11RQ12RQ13RQ14RQ1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18REEvaluate the integral. 19. x+19x2+6x+5dx20RE21RE22RE23RE24RE25REEvaluate the integral. 26. xsinxcosxdx27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66REThe speedometer reading (v) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpsons Rule to estimate the distance traveled by the car.A population of honeybees increased at a rate of r(t) bees per week, where the graph of r is as shown. Use Simpsons Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks.69RESuppose you are asked to estimate the volume of a football. You measure and find that a football is 28 cm long. You use a piece of string and measure the circumference at its widest point to be 53 cm. The circumference 7 cm from each end is 45 cm. Use Simpsons Rule to make your estimate.71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE1P2P3P4P5P6P7P8P9P10P11P12P13P14P15P16PEvaluate the integral using integration by parts with the indicated choices of u and dv. 1. xe2xdx;u=x,dv=e2xdx2EEvaluate the integral. 3. xcos5xdxEvaluate the integral. 4. ye0.2ydyEvaluate the integral. 5. te3tdt6E7EEvaluate the integral. 8. t2sintdtEvaluate the integral. 9. cos1xdx10E11EEvaluate the integral. 12. tan12ydy13E14EEvaluate the integral. 15. (lnx)2dxEvaluate the integral. 16. z10zdz17E18E19EEvaluate the integral. 20. xtan2xdx21E22EEvaluate the integral. 23. 01/2xcosxdxEvaluate the integral. 24. 01(x2+1)exdxEvaluate the integral. 25. 02ysinhydyEvaluate the integral. 26. 12w2lnwdwEvaluate the integral. 27. 15lnRR2dREvaluate the integral. 28. 02t2sin2tdtEvaluate the integral. 29. 0xsinxcosxdxEvaluate the integral. 30. 13arctan(1/x)dxEvaluate the integral. 31. 15MeMdMEvaluate the integral. 32. 12(lnx)2x3dxEvaluate the integral. 33. 0/3sinxln(cosx)dxEvaluate the integral. 34. 01r34+r2drEvaluate the integral. 35. 12x4(lnx)2dxEvaluate the integral. 36. 0tessin(ts)dsFirst make a substitution and then use integration by parts to evaluate the integral. 37. exdxFirst make a substitution and then use integration by parts to evaluate the integral. 38. cos(lnx)dxFirst make a substitution and then use integration by parts to evaluate the integral. 39. /23cos(2)dFirst make a substitution and then use integration by parts to evaluate the integral. 40. 0ecostsin2tdtFirst make a substitution and then use integration by parts to evaluate the integral. 41. xln(1+x)dxFirst make a substitution and then use integration by parts to evaluate the integral. 42. arcsin(lnx)xdxEvaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). 43. xe2xdxEvaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). 44. x3/2lnxdx45E46E47E(a) Prove the reduction formula cosnxdx=1ncosn1xsinx+n1ncosn2xdx (b) Use part (a) to evaluate cos2xdx. (c) Use parts (a) and (b) to evaluate cos4xdx.49E50E51EUse integration by parts to prove the reduction formula. 52. xnexdx=xnexnxn1exdx53E54E55E56E57EFind the area of the region bounded by the given curves. 58.y = x2 ex, y = xex59E60E61EUse the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. 62.y = ex, y = ex, x = 1; about the y-axis63E64ECalculate the volume generated by rotating the region bounded by the curves y = ln x, y = 0, and x = 2 about each axis. (a) The y-axis (b) The x-axisCalculate the average value of f(x) = x sec2x on the interval [0, /4].67EA rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is m, the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity ve (relative to the rocket). A model for the velocity of the rocket at time t is given by the equation v(t)=gtvelnmrtm where g is the acceleration due to gravity and t is not too large. If g = 9.8 m/s2, m = 30,000 kg, r = 160 kg/s, and ve = 3000 m/s, find the height of the rocket one minute after liftoff.A particle that moves along a straight line has velocity v(t) = t2et meters per second after t seconds. How far will it travel during the first t seconds?70ESuppose that f(1)=2,f(4)=7,f(1)=5,f(4)=3 and f" is continuous. Find the value of 14xf(x)dx.(a)Use integration by parts to show that f(x)dx=xf(x)xf(x)dx (b) If f and g are inverse functions and f' is continuous, prove that abf(x)dx=bf(b)af(a)f(a)f(b)g(y)dy [Hint: Use part (a) and make the substitution y = y(x).] In the case where f and g are positive functions and b a 0, draw a diagram to give a geometric interpretation of part (b). (d) Use part (b) to evaluate 1elnxdx.73ELet In=0/2sinnxdx (a) Show that I2n+2I2n+2I2n. (b) Use Exercise 50 to show that I2n+2I2n=2n+12n+2 (c) Use parts (a) and (b) to show that 2n+12n+2I2n+1I2n+11 and deduce that limnI2n+1/I2n=1. (d) Use part (c) and Exercises 49 and 50 to show that limn2123434565672n2n12n2n+1=2 This formula is usually written as an infinite product: 2=212343456567 and is called the Wallis product. (e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle (see the figure). Find the limit of the ratios of width to height of these rectangles.Evaluate the integral. 1. sin2xcos3xdxEvaluate the integral. 2. sin3cos4dEvaluate the integral. 3. 0/2sin7cos5d4EEvaluate the integral. 5. sin5(2t)cos2(2t)dtEvaluate the integral. 6. tcos5(t2)dt7EEvaluate the integral. 8. 02sin2(13)dEvaluate the integral. 9. 0cos4(2t)dtEvaluate the integral. 10. 0sin2tcos4tdt11E12E13EEvaluate the integral. 14. sin2(1/t)t2dt15E16E17EEvaluate the integral. 18. sinxcos(12x)dx19EEvaluate the integral. 20. xsin3xdx21EEvaluate the integral. 22. tan2sec4d23E24EEvaluate the integral. 25. tan4xsec6xdx26E27EEvaluate the integral. 28. tan2xsec3xdx29E30E31E32EEvaluate the integral. 33. xsecxtanxdxEvaluate the integral. 34. sincos3d35E36EEvaluate the integral. 37. /4/2cot5csc3dEvaluate the integral. 38. /4/2csc4cot4d39E40E41EEvaluate the integral. 42. sin2sin6d43EEvaluate the integral. 44. sinxsec5xdx45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61EFind the volume obtained by rotating the region bounded by the curves about the given axis. 62.y=sin2x,y=0,0x; about the x-axisFind the volume obtained by rotating the region bounded by the curves about the given axis. 63.y=sinx,y=cosx,0x/4; about y = 164E65E66E67EProve the formula, where m and n are positive integers. 68. sinmxcosnxdx={0ifmnifm=n69E70EEvaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. 1. dxx24x2x=2sinEvaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. 2. x3x2+4dxx=2tan3E4E