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All Textbook Solutions for Single Variable Calculus: Early Transcendentals
Evaluate the integral. 5. x21x4dxEvaluate the integral. 6. 03x36x2dx7E8E9E10E11EEvaluate the integral. 12. 02dt4+t213E14E15E16E17E18E19E20E21E22EEvaluate the integral. 23. dxx2+2x+524E25E26EEvaluate the integral. 27. x2+2xdx28E29E30E31E32E33E34E35E36E37EFind the volume of the solid obtained by rotating about the line x = 1 the region under the curve y=x1x2, 0 x 1.(a) Use trigonometric substitution to verify that 0xa2t2dt=12a2sin1(x/a)+12xa2x2 (b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a). y=a2t240EA torus is generated by rotating the circle x2 + (y R)2 = r2 about the x-axis. Find the volume enclosed by the torus.A charged rod of length L produces an electric field at point P(a, b) given by E(P)=aLab40(x2+b2)3/2dx where is the charge density per unit length on the rod and 0 is the free space permittivity (see the figure). Evaluate the integral to determine an expression for the electric field E(P).43E44EWrite out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 1. (a) 4+x(1+2x)(3x) (b) 1xx3+x4Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 2. (a) x6x2+x6(b) x2x2+x+63E4EWrite out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 5. (a) x6x24(b) x4(x2x+1)(x2+2)2Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 6. (a) t6+1t6+t3(b) x5+1(x2+x)(x4+2x2+1)Evaluate the integral. 7. x4x1dx8E9EEvaluate the integral. 10. y(y+4)(2y1)dy11EEvaluate the integral. 12. 01x4x25x+6dx13E14E15E16E17E18EEvaluate the integral. 19. 01x2+x+1(x+1)2(x+2)dx20E21E22E23E24E25E26E27EEvaluate the integral. 28. x3+6x2x4+6x2dxEvaluate the integral. 29. x+4x2+2x+5dx30E31E32E33E34E35E36E37EEvaluate the integral. 38. x3+2x2+3x2(x2+2x+2)2dx39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54EUse a graph of f(x) = 1/(x2 2x 3) to decide whether 02f(x) is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the exact value.Evaluate 1x2+kdx by considering several cases for the constant k.57E58EThe German mathematician Karl Weierstrass (18151897) noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function of t. (a) If t = tan(x/2), x , sketch a right triangle or use trigonometric identities to show that cos(x2)=11+t2andsin(x2)=t1+t2 (b) Show that cosx=1t21+t2andsinx=2t1+t2 (c) Show that dx=21+t2dt60E61E62E63E64E65E66E67E68E69E70E71E(a) Use integration by parts to show that, for any positive integer n, dx(x2+a2)n=x2a2(n1)(x2+a2)n1+2n32a2(n1)dx(x2+a2)n1 (b) Use part (a) to evaluate dx(x2+1)2anddx(x2+1)373EIf f is a quadratic function such that f(0) = 1 and f(x)x2(x+1)3dx is a rational function, find the value of f(0).75EEvaluate the integral. 1. cosx1sinxdx2E3EEvaluate the integral. 4. sin3xcosxdx5E6EEvaluate the integral. 7. 11earctany1+y2dy8E9EEvaluate the integral. 10. cos(1/x)x3dx11E12E13E14EEvaluate the integral. 15. xsecxtanxdx16E17E18E19E20E21EEvaluate the integral. 22. lnxx1+(lnx)2dx23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78EEvaluate the integral. 79. xsin2xcosxdx80E81E82E83E84EUse the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. 1. 0/2cos5xcos2xdx;entry802E3E4E5EUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 6. 02x24x2dx7E8E9EUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 10. 2y23y2dy11E12EUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 13. arctanxxdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 14. 0x3sinxdx15E16E17EUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 18. dx2x33x219E20E21EUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 22. 02x34x2x4dx23E24E25EUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 26. 01x4exdx27E28E29EUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 30. etsin(t3)dt31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46ELet l=04f(x)dx where f is the function whose graph is shown. (a) Use the graph to find L2, R2, and M2. (b) Are these underestimates or overestimates of l? (c) Use the graph to find T2 How does it compare with l? (d) For any value of n, list the numbers Ln, Rn, Mn, Tn,. and l in increasing order.2E3E4E5E6EUse (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 7. 12x31dx,n=108EUse (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 9. 02ex1+x2dx,n=10Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 10. 0/21+cosx3dx,n=411E12E13E14E15E16E17E18E(a) Find the approximations T8 and M8 for the integral 01cos(x2)dx. (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose n so that the approximations Tn and Mn to the integral in part (a) are accurate to within 0.0001?20E21EHow large should n be to guarantee that the Simpsons Rule approximation to 01ex2dx is accurate to within 0.00001?23E24E25E26E27EFind the approximations Tn, Mn, and Sn for n = 6 and 12. Then compute the corresponding errors ET, EM, and Es. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? In particular, what happens to the errors when n is doubled? 28. 141xdx29EThe widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpsons Rule to estimate the area of the pool.31E32EA graph of the temperature in Boston on August 11, 2013, is shown. Use Simpsons Rule with n = 12 to estimate the average temperature on that day.34EThe graph of the acceleration a(t) of a car measured in ft/s2 is shown. Use Simpsons Rule to estimate the increase in the velocity of the car during the 6-second time interval.Water leaked from a tank at a rate of r(t) liters per hour, where the graph of r is as shown. Use Simpsons Rule to estimate the total amount of water that leaked out during the first 6 hours.37E38E39E40EThe region bounded by the curve y=1/(1+ex), the x-and y-axes, and the line x = 10 is rotated about the x-axis. Use Simpsons Rule with n = 10 to estimate the volume of the resulting solid.The figure shows a pendulum with length L that makes a maximum angle 0 with the vertical. Using Newtons Second Law, it can be shown that the period T (the time for one complete swing) is given by T=4Lg0/2dx1k2sin2x where k=sin(120) and g is the acceleration due to gravity. If L = 1 m and 0 = 42, use Simpsons Rule with n = 10 to find the period.43E44E45E46E47E48E49EShow that 13Tn+23Mn=S2n.Explain why each of the following integrals is improper. (a) 12xx1dx (b) 011+x3dx (c) 0x2ex2dx (d) 0/4cotxdx2E3E4EDetermine whether each integral is convergent or divergent. Evaluate those that are convergent. 5. 31(x2)3/2dx