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Math Calculus Single Variable Essential Calculus: Early Transcendentals

(a) Find the approximations T 10 , M 10 , and S 10 for ∫ 0 π sin x d x and the corresponding errors E T , E M , and E S . (b) Compare the actual errors in part (a) with the error estimates given by (3) and (4). (c) How large do we have to choose n so that the approximations T n , M n , and S n . to the integral in part (a) are accurate to within 0.00001?

(a) Find the approximations T 10 , M 10 , and S 10 for ∫ 0 π sin x d x and the corresponding errors E T , E M , and E S . (b) Compare the actual errors in part (a) with the error estimates given by (3) and (4). (c) How large do we have to choose n so that the approximations T n , M n , and S n . to the integral in part (a) are accurate to within 0.00001?

Single Variable Essential Calculus: Early Transcendentals

Single Variable Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112785

Author: James Stewart

Publisher: Cengage Learning

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Textbook Question

Chapter 6.5, Problem 19E

(a) Find the approximations T₁₀, M₁₀, and S₁₀ for ∫ 0 π sin x d x and the corresponding errors E_T, E_M, and E_S.

(b) Compare the actual errors in part (a) with the error estimates given by (3) and (4).

(c) How large do we have to choose n so that the approximations T_n, M_n, and S_n. to the integral in part (a) are accurate to within 0.00001?

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Chapter 6.5, Problem 18E

Chapter 6.5, Problem 20E

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Chapter 6 Solutions

Single Variable Essential Calculus: Early Transcendentals

Ch. 6.1 - Evaluate the integral using integration by parts...Ch. 6.1 - Evaluate the integral using integration by parts...Ch. 6.1 - Evaluate the integral. 3. xcos5xdx Ch. 6.1 - Evaluate the integral. 4. ye0.2ydy Ch. 6.1 - Evaluate the integral. 5. te3tdt Ch. 6.1 - Evaluate the integral. 6. (x1)sinxdx Ch. 6.1 - Evaluate the integral. 7. (x2+2x)cosxdx Ch. 6.1 - Evaluate the integral. 8. t2sintdt Ch. 6.1 - Evaluate the integral. ln(2x + 1) dx Ch. 6.1 - Evaluate the integral. p5lnpdp

Ch. 6.1 - Prob. 11E Ch. 6.1 - Evaluate the integral. sin1xdx Ch. 6.1 - Evaluate the integral. 17. e2sin3d Ch. 6.1 - Evaluate the integral. 18. ecos2d Ch. 6.1 - Evaluate the integral. t3etdt Ch. 6.1 - Evaluate the integral. 21. xe2x(1+2x)2dx Ch. 6.1 - Evaluate the integral. 23. 01/2xcosxdx Ch. 6.1 - Prob. 18E Ch. 6.1 - Evaluate the integral. 49lnyydy Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 19E Ch. 6.1 - Evaluate the integral. 01tcoshtdt Ch. 6.1 - Prob. 23E Ch. 6.1 - Evaluate the integral. 34. 01r34+r2dr Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 31E Ch. 6.1 - (a) Prove the reduction formula...Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - A rocket accelerates by burning its onboard fuel,...Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - (a) Use integration by parts to show that...Ch. 6.2 - Evaluate the integral. 1. sin2xcos3xdx Ch. 6.2 - Evaluate the integral. 2. sin3cos4d Ch. 6.2 - Evaluate the integral. 3. 0/2sin7cos5d Ch. 6.2 - Prob. 4E Ch. 6.2 - Prob. 5E Ch. 6.2 - Prob. 6E Ch. 6.2 - Prob. 7E Ch. 6.2 - Evaluate the integral. 11. 0/2sin2xcos2xdx Ch. 6.2 - Prob. 8E Ch. 6.2 - Evaluate the integral. 0cos6d Ch. 6.2 - Evaluate the integral. t sin2t dt Ch. 6.2 - Prob. 12E Ch. 6.2 - Evaluate the integral. cos2x tan3x dx Ch. 6.2 - Prob. 14E Ch. 6.2 - Evaluate the integral. 1sinxcosxdx Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Prob. 18E Ch. 6.2 - Evaluate the integral. 23. tan2xdx Ch. 6.2 - Evaluate the integral. 24. (tan2x+tan4x)dx Ch. 6.2 - Evaluate the integral. 25. tan4xsec6xdx Ch. 6.2 - Prob. 22E Ch. 6.2 - Evaluate the integral. 27. tan3xsecxdx Ch. 6.2 - Evaluate the integral. 28. tan5xsec3xdx Ch. 6.2 - Prob. 23E Ch. 6.2 - Evaluate the integral. 30. 0/4tan3tdt Ch. 6.2 - Evaluate the integral. 31. tan5xdx Ch. 6.2 - Prob. 28E Ch. 6.2 - Prob. 29E Ch. 6.2 - Prob. 30E Ch. 6.2 - Prob. 31E Ch. 6.2 - Evaluate the integral. csc4x cot6x dx Ch. 6.2 - Prob. 33E Ch. 6.2 - Prob. 35E Ch. 6.2 - Prob. 34E Ch. 6.2 - Prob. 36E Ch. 6.2 - Prob. 37E Ch. 6.2 - Prob. 38E Ch. 6.2 - Prob. 65E Ch. 6.2 - Prob. 66E Ch. 6.2 - Prob. 39E Ch. 6.2 - Prob. 40E Ch. 6.2 - Evaluate the integral using the indicated...Ch. 6.2 - Prob. 42E Ch. 6.2 - Prob. 43E Ch. 6.2 - Prob. 44E Ch. 6.2 - Evaluate the integral. 7. 0adx(a2+x2)3/2, a 0 Ch. 6.2 - Prob. 46E Ch. 6.2 - Prob. 47E Ch. 6.2 - Prob. 48E Ch. 6.2 - Prob. 49E Ch. 6.2 - Prob. 50E Ch. 6.2 - Prob. 51E Ch. 6.2 - Prob. 58E Ch. 6.2 - Prob. 52E Ch. 6.2 - Prob. 55E Ch. 6.2 - Prob. 54E Ch. 6.2 - Prob. 57E Ch. 6.2 - Prob. 61E Ch. 6.2 - Prob. 53E Ch. 6.2 - Prob. 56E Ch. 6.2 - Prob. 62E Ch. 6.2 - Prob. 63E Ch. 6.2 - Prob. 64E Ch. 6.2 - Prob. 59E Ch. 6.2 - Evaluate the integral. 30. 0/2cost1+sin2tdt Ch. 6.2 - Prob. 67E Ch. 6.2 - Find the area of the region bounded by the...Ch. 6.2 - Prob. 69E Ch. 6.2 - Prob. 70E Ch. 6.3 - Write out the form of the partial fraction...Ch. 6.3 - Write out the form of the partial fraction...Ch. 6.3 - Write out the form of the partial fraction...Ch. 6.3 - Write out the form of the partial fraction...Ch. 6.3 - Write out the form of the partial fraction...Ch. 6.3 - Write out the form of the partial fraction...Ch. 6.3 - Evaluate the integral. 7. x4x1dx Ch. 6.3 - Evaluate the integral. 8. 3t2t+1dt Ch. 6.3 - Evaluate the integral. 9. 5x+1(2x+1)(x1)dx Ch. 6.3 - Evaluate the integral. 10. y(y+4)(2y1)dy Ch. 6.3 - Evaluate the integral. 11. 0122x2+3x+1dx Ch. 6.3 - Evaluate the integral. 12. 01x4x25x+6dx Ch. 6.3 - 41550-7.4-13E 7–38. Evaluate the integral. 13. Ch. 6.3 - Evaluate the integral. 14. 1(x+a)(x+b)dx Ch. 6.3 - Prob. 15E Ch. 6.3 - Evaluate the integral. 01x34x10x2x6dx Ch. 6.3 - Prob. 17E Ch. 6.3 - Evaluate the integral. x2+2x1x3xdx Ch. 6.3 - Evaluate the integral. x2+2x1x3xdx Ch. 6.3 - Evaluate the integral. x25x+16(2x+1)(x2)2dx Ch. 6.3 - Evaluate the integral. x3+4x2+4dx Ch. 6.3 - Evaluate the integral. x22x1(x1)2(x2+1)dx Ch. 6.3 - Evaluate the integral. 23. 10(x1)(x2+9)dx Ch. 6.3 - Evaluate the integral. 24. x2x+6x3+3xdx Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 34E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 32E Ch. 6.3 - Make a substitution to express the integrand as a...Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.3 - Make a substitution to express the integrand as a...Ch. 6.3 - Prob. 39E Ch. 6.3 - Prob. 40E Ch. 6.3 - Prob. 41E Ch. 6.3 - Prob. 42E Ch. 6.3 - One method of slowing the growth of an insect...Ch. 6.3 - 41550-7.4-68E 68. Factor x4 + 1 as a difference of...Ch. 6.3 - Suppose that F, G, and Q are polynomials and...Ch. 6.3 - If f is a quadratic function such that f(0) = 1...Ch. 6.3 - Prob. 47E Ch. 6.4 - Use the Table of Integrals on Reference Pages 610...Ch. 6.4 - Use the Table of Integrals on Reference Pages 610...Ch. 6.4 - Prob. 3E Ch. 6.4 - Prob. 4E Ch. 6.4 - Prob. 5E Ch. 6.4 - Use the Table of Integrals on Reference Pages 610...Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 9E Ch. 6.4 - Prob. 10E Ch. 6.4 - Prob. 11E Ch. 6.4 - Prob. 8E Ch. 6.4 - Prob. 13E Ch. 6.4 - Prob. 12E Ch. 6.4 - Prob. 15E Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 19E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 14E Ch. 6.4 - Prob. 21E Ch. 6.4 - Prob. 22E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 20E Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.5 - Let I=04f(x)dx, where f is the function whose...Ch. 6.5 - Prob. 2E Ch. 6.5 - Estimate 01cos(x2)dx using (a) the Trapezoidal...Ch. 6.5 - Prob. 4E Ch. 6.5 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 6.5 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 6.5 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 6.5 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 6.5 - Use (a) the Trapezoidal Rule, (b) the Midpoint...Ch. 6.5 - 41550-7.7-10E 7–18. Use (a) the Trapezoidal Rule,...Ch. 6.5 - Prob. 11E Ch. 6.5 - Prob. 12E Ch. 6.5 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 6.5 - Prob. 14E Ch. 6.5 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 6.5 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 6.5 - Prob. 17E Ch. 6.5 - Prob. 18E Ch. 6.5 - 41550-7.7-21E 21. (a) Find the approximations T10,...Ch. 6.5 - How large should n be to guarantee that the...Ch. 6.5 - Find the approximations Ln, Rn, Tn, and Mn to the...Ch. 6.5 - Find the approximations Tn, Mn, and Sn. for n = 6...Ch. 6.5 - Estimate the area under the graph in the figure by...Ch. 6.5 - A radar gun was used to record the speed of a...Ch. 6.5 - The graph of the acceleration a(t) of a car...Ch. 6.5 - Water leaked from a tank at a rate of r(t) liters...Ch. 6.5 - A graph of the temperature in New York City on...Ch. 6.5 - Prob. 30E Ch. 6.5 - (a) Use the Midpoint Rule and the given data to...Ch. 6.5 - The table (supplied by San Diego Gas and Electric)...Ch. 6.5 - Shown is the graph of traffic on an Internet...Ch. 6.5 - The figure shows a pendulum with length L that...Ch. 6.5 - The intensity of light with wavelength traveling...Ch. 6.5 - Prob. 38E Ch. 6.5 - Prob. 36E Ch. 6.5 - Prob. 37E Ch. 6.5 - Prob. 39E Ch. 6.5 - Prob. 40E Ch. 6.5 - Show that 12(Tn+Mn)=T2n.Ch. 6.5 - Show that 13Tn+23Mn=S2n.Ch. 6.6 - Explain why each of the following integrals is...Ch. 6.6 - Which of the following integrals are improper?...Ch. 6.6 - 41550-7.8-3E 3. Find the area under the curve y =...Ch. 6.6 - Prob. 4E Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Prob. 7E Ch. 6.6 - Prob. 8E Ch. 6.6 - Prob. 9E Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - 41550-7.8-29E 5-40 Determine whether each integral...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Sketch the region and find its area (if the area...Ch. 6.6 - Sketch the region and find its area (if the area...Ch. 6.6 - Sketch the region and find its area (if the area...Ch. 6.6 - Sketch the region and find its area (if the area...Ch. 6.6 - Sketch the region and find its area (if the area...Ch. 6.6 - Sketch the region and find its area (if the area...Ch. 6.6 - (a) If g(x) = (sin2x)/x2, use your calculator or...Ch. 6.6 - (a) If g(x)=1/(x1), use your calculator or...Ch. 6.6 - Use the Comparison Theorem to determine whether...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Use the Comparison Theorem to determine whether...Ch. 6.6 - Use the Comparison Theorem to determine whether...Ch. 6.6 - 41550-7.8-53E 49–54 Use the Comparison Theorem to...Ch. 6.6 - Use the Comparison Theorem to determine whether...Ch. 6.6 - 41550-7.8-55E 55. The integral is improper for...Ch. 6.6 - Find the values of p for which the integral...Ch. 6.6 - Find the values of p for which the integral...Ch. 6.6 - 41550-7.8-60E 60. (a) Evaluate the integral for n...Ch. 6.6 - 41550-7.8-61E 61. (a) Show that is divergent. (b)...Ch. 6.6 - 41550-7.8-62E 62. The average speed of molecules...Ch. 6.6 - Astronomers use a technique called stellar...Ch. 6.6 - 41550-7.8-67E 67. A manufacturer of lightbulbs...Ch. 6.6 - As we saw in Section 3.4, a radioactive substance...Ch. 6.6 - Determine how large the number a has to be so that...Ch. 6.6 - Estimate the numerical value of 0ex2dx by writing...Ch. 6.6 - 41550-7.8-76E 76. If is convergent and a and b...Ch. 6.6 - Show that 0x2ex2dx=120ex2dx.Ch. 6.6 - 41550-7.8-78E 78. Show that by interpreting the...Ch. 6.6 - Find the value of the constant C for which the...Ch. 6.6 - Find the value of the constant C for which the...Ch. 6.6 - Suppose f is continuous on [0, ) and limxf(x) = 1....Ch. 6.6 - Show that if a 1 and b a + 1, then the...Ch. 6 - Prob. 1RCC Ch. 6 - How do you evaluate sinmxcosnxdx if m is odd? What...Ch. 6 - Prob. 3RCC Ch. 6 - Prob. 4RCC Ch. 6 - Prob. 5RCC Ch. 6 - Prob. 6RCC Ch. 6 - Prob. 7RCC Ch. 6 - Prob. 8RCC Ch. 6 - Prob. 1RQ Ch. 6 - Prob. 2RQ Ch. 6 - Prob. 3RQ Ch. 6 - Prob. 4RQ Ch. 6 - Prob. 5RQ Ch. 6 - Prob. 6RQ Ch. 6 - Prob. 7RQ Ch. 6 - Prob. 8RQ Ch. 6 - Prob. 9RQ Ch. 6 - 41550-7-10RQ Determine whether the statement is...Ch. 6 - 41550-7-11RQ Determine whether the statement is...Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Evaluate the integral. 1. 12(x+1)2xdx Ch. 6 - Evaluate the integral. 2. 12x(x+1)2dx Ch. 6 - Evaluate the integral. 0/2sinecosd Ch. 6 - 41550-7-4RE 1–40 Evaluate the integral. 4. Ch. 6 - Evaluate the integral. 5. dt2t2+3t+1 Ch. 6 - Prob. 6RE Ch. 6 - Prob. 15RE Ch. 6 - Prob. 8RE Ch. 6 - Prob. 7RE Ch. 6 - Evaluate the integral. 10. 01arctanx1+x2dx Ch. 6 - Prob. 11RE Ch. 6 - Prob. 12RE Ch. 6 - Prob. 13RE Ch. 6 - Prob. 14RE Ch. 6 - Evaluate the integral. 14x3/2lnxdx Ch. 6 - Evaluate the integral. 16. sec6tan2d Ch. 6 - Prob. 17RE Ch. 6 - Prob. 18RE Ch. 6 - Prob. 19RE Ch. 6 - Prob. 26RE Ch. 6 - Prob. 21RE Ch. 6 - Prob. 22RE Ch. 6 - Prob. 23RE Ch. 6 - Evaluate the integral. 24. excosxdx Ch. 6 - Evaluate the integral. 25. 3x3x2+6x4(x2+1)(x2+2)dx Ch. 6 - Prob. 20RE Ch. 6 - Prob. 27RE Ch. 6 - Prob. 28RE Ch. 6 - Prob. 29RE Ch. 6 - Prob. 30RE Ch. 6 - Prob. 31RE Ch. 6 - Prob. 32RE Ch. 6 - Prob. 33RE Ch. 6 - Prob. 34RE Ch. 6 - Prob. 35RE Ch. 6 - Prob. 36RE Ch. 6 - Prob. 37RE Ch. 6 - Prob. 38RE Ch. 6 - Prob. 39RE Ch. 6 - Prob. 40RE Ch. 6 - Prob. 41RE Ch. 6 - Prob. 42RE Ch. 6 - Evaluate the integral or show that it is...Ch. 6 - Prob. 44RE Ch. 6 - Prob. 45RE Ch. 6 - Prob. 46RE Ch. 6 - Prob. 47RE Ch. 6 - Prob. 48RE Ch. 6 - Prob. 49RE Ch. 6 - Prob. 50RE Ch. 6 - Prob. 51RE Ch. 6 - Prob. 52RE Ch. 6 - Prob. 53RE Ch. 6 - Prob. 54RE Ch. 6 - Prob. 55RE Ch. 6 - Prob. 56RE Ch. 6 - Prob. 57RE Ch. 6 - Prob. 58RE Ch. 6 - Prob. 59RE Ch. 6 - Use Simpsons Rule with n = 6 to estimate the area...Ch. 6 - The speedometer reading (v) on a car was observed...Ch. 6 - Prob. 62RE Ch. 6 - Prob. 63RE Ch. 6 - Prob. 64RE Ch. 6 - Prob. 65RE

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