Essential Calculus
2nd Edition
ISBN: 9781133490975
Author: Stewart, James
Publisher: Cengage Learning
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Textbook Question
Chapter 6.5, Problem 1E
Let
(a) Use the graph to find L2, R2, and M2.
(b) Are these underestimates or overestimates of I?
(c) Use the graph to find T2. How does it compare with I?
(d) For any value of n, list the numbers Ln, Rn, Mn, Tn, and I in increasing order.
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Please solve 14 and 15
1. Consider the following system of equations:
x13x2 + 4x3 - 5x4 = 7
-2x13x2 + x3 - 6x4 = 7
x16x213x3 - 21x4 = 28
a) Solve the system. Write your solution in parametric and vector form.
b) What is a geometric description of the solution.
7
c) Is v =
7 in the span of the set S=
[28.
1
HE
3
-5
3
·6
? If it is, write v
6
as a linear combination of the vectors in S. Justify.
d) How many solutions are there to the associated homogeneous system for
the system above? Justify.
e) Let A be the coefficient matrix from the system above. Find the set of all
solutions to Ax = 0.
f) Is there a solution to Ax=b for all b in R³? Justify.
Chapter 6 Solutions
Essential Calculus
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. xcos5xdxCh. 6.1 - Evaluate the integral. 4. ye0.2ydyCh. 6.1 - Evaluate the integral. 5. te3tdtCh. 6.1 - Evaluate the integral. 6. (x1)sinxdxCh. 6.1 - Evaluate the integral. 7. (x2+2x)cosxdxCh. 6.1 - Evaluate the integral. 8. t2sintdtCh. 6.1 - Evaluate the integral. ln(2x + 1) dxCh. 6.1 - Evaluate the integral. p5lnpdp
Ch. 6.1 - Prob. 11ECh. 6.1 - Evaluate the integral. sin1xdxCh. 6.1 - Evaluate the integral. 17. e2sin3dCh. 6.1 - Evaluate the integral. 18. ecos2dCh. 6.1 - Evaluate the integral. t3etdtCh. 6.1 - Evaluate the integral. 21. xe2x(1+2x)2dxCh. 6.1 - Evaluate the integral. 23. 01/2xcosxdxCh. 6.1 - Prob. 18ECh. 6.1 - Evaluate the integral. 49lnyydyCh. 6.1 - Prob. 22ECh. 6.1 - Prob. 19ECh. 6.1 - Evaluate the integral. 01tcoshtdtCh. 6.1 - Prob. 23ECh. 6.1 - Evaluate the integral. 34. 01r34+r2drCh. 6.1 - Prob. 25ECh. 6.1 - Prob. 26ECh. 6.1 - Prob. 30ECh. 6.1 - Prob. 27ECh. 6.1 - Prob. 29ECh. 6.1 - Prob. 28ECh. 6.1 - Prob. 31ECh. 6.1 - (a) Prove the reduction formula...Ch. 6.1 - Prob. 33ECh. 6.1 - Prob. 34ECh. 6.1 - Prob. 35ECh. 6.1 - Prob. 36ECh. 6.1 - Prob. 37ECh. 6.1 - Prob. 38ECh. 6.1 - Prob. 39ECh. 6.1 - Prob. 40ECh. 6.1 - Prob. 41ECh. 6.1 - A rocket accelerates by burning its onboard fuel,...Ch. 6.1 - Prob. 43ECh. 6.1 - Prob. 44ECh. 6.1 - Prob. 45ECh. 6.1 - (a) Use integration by parts to show that...Ch. 6.2 - Evaluate the integral. 1. sin2xcos3xdxCh. 6.2 - Evaluate the integral. 2. sin3cos4dCh. 6.2 - Evaluate the integral. 3. 0/2sin7cos5dCh. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.2 - Prob. 7ECh. 6.2 - Evaluate the integral. 11. 0/2sin2xcos2xdxCh. 6.2 - Prob. 8ECh. 6.2 - Evaluate the integral. 0cos6dCh. 6.2 - Evaluate the integral. t sin2t dtCh. 6.2 - Prob. 12ECh. 6.2 - Evaluate the integral. cos2x tan3x dxCh. 6.2 - Prob. 14ECh. 6.2 - Evaluate the integral. 1sinxcosxdxCh. 6.2 - Prob. 16ECh. 6.2 - Prob. 17ECh. 6.2 - Prob. 18ECh. 6.2 - Evaluate the integral. 23. tan2xdxCh. 6.2 - Evaluate the integral. 24. (tan2x+tan4x)dxCh. 6.2 - Evaluate the integral. 25. tan4xsec6xdxCh. 6.2 - Prob. 22ECh. 6.2 - Evaluate the integral. 27. tan3xsecxdxCh. 6.2 - Evaluate the integral. 28. tan5xsec3xdxCh. 6.2 - Prob. 23ECh. 6.2 - Evaluate the integral. 30. 0/4tan3tdtCh. 6.2 - Evaluate the integral. 31. tan5xdxCh. 6.2 - Prob. 28ECh. 6.2 - Prob. 29ECh. 6.2 - Prob. 30ECh. 6.2 - Prob. 31ECh. 6.2 - Evaluate the integral. csc4x cot6x dxCh. 6.2 - Prob. 33ECh. 6.2 - Prob. 35ECh. 6.2 - Prob. 34ECh. 6.2 - Prob. 36ECh. 6.2 - Prob. 37ECh. 6.2 - Prob. 38ECh. 6.2 - Prob. 65ECh. 6.2 - Prob. 66ECh. 6.2 - Prob. 39ECh. 6.2 - Prob. 40ECh. 6.2 - Evaluate the integral using the indicated...Ch. 6.2 - Prob. 42ECh. 6.2 - Prob. 43ECh. 6.2 - Prob. 44ECh. 6.2 - Evaluate the integral. 7. 0adx(a2+x2)3/2, a 0Ch. 6.2 - Prob. 46ECh. 6.2 - Prob. 47ECh. 6.2 - Prob. 48ECh. 6.2 - Prob. 49ECh. 6.2 - Prob. 50ECh. 6.2 - Prob. 51ECh. 6.2 - Prob. 58ECh. 6.2 - Prob. 52ECh. 6.2 - Prob. 55ECh. 6.2 - Prob. 54ECh. 6.2 - Prob. 57ECh. 6.2 - Prob. 61ECh. 6.2 - Prob. 53ECh. 6.2 - Prob. 56ECh. 6.2 - Prob. 62ECh. 6.2 - Prob. 63ECh. 6.2 - Prob. 64ECh. 6.2 - Prob. 59ECh. 6.2 - Evaluate the integral. 30. 0/2cost1+sin2tdtCh. 6.2 - Prob. 67ECh. 6.2 - Find the area of the region bounded by the...Ch. 6.2 - Prob. 69ECh. 6.2 - Prob. 70ECh. 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. x4x1dxCh. 6.3 - Evaluate the integral. 8. 3t2t+1dtCh. 6.3 - Evaluate the integral. 9. 5x+1(2x+1)(x1)dxCh. 6.3 - Evaluate the integral. 10. y(y+4)(2y1)dyCh. 6.3 - Evaluate the integral. 11. 0122x2+3x+1dxCh. 6.3 - Evaluate the integral. 12. 01x4x25x+6dxCh. 6.3 - 41550-7.4-13E
7–38. Evaluate the integral.
13.
Ch. 6.3 - Evaluate the integral. 14. 1(x+a)(x+b)dxCh. 6.3 - Prob. 15ECh. 6.3 - Evaluate the integral. 01x34x10x2x6dxCh. 6.3 - Prob. 17ECh. 6.3 - Evaluate the integral. x2+2x1x3xdxCh. 6.3 - Evaluate the integral. x2+2x1x3xdxCh. 6.3 - Evaluate the integral. x25x+16(2x+1)(x2)2dxCh. 6.3 - Evaluate the integral. x3+4x2+4dxCh. 6.3 - Evaluate the integral. x22x1(x1)2(x2+1)dxCh. 6.3 - Evaluate the integral. 23. 10(x1)(x2+9)dxCh. 6.3 - Evaluate the integral. 24. x2x+6x3+3xdxCh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Prob. 27ECh. 6.3 - Prob. 28ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 29ECh. 6.3 - Prob. 33ECh. 6.3 - Prob. 34ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 32ECh. 6.3 - Make a substitution to express the integrand as a...Ch. 6.3 - Prob. 36ECh. 6.3 - Prob. 37ECh. 6.3 - Make a substitution to express the integrand as a...Ch. 6.3 - Prob. 39ECh. 6.3 - Prob. 40ECh. 6.3 - Prob. 41ECh. 6.3 - Prob. 42ECh. 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. 47ECh. 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. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.4 - Use the Table of Integrals on Reference Pages 610...Ch. 6.4 - Prob. 7ECh. 6.4 - Prob. 9ECh. 6.4 - Prob. 10ECh. 6.4 - Prob. 11ECh. 6.4 - Prob. 8ECh. 6.4 - Prob. 13ECh. 6.4 - Prob. 12ECh. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Prob. 19ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 14ECh. 6.4 - Prob. 21ECh. 6.4 - Prob. 22ECh. 6.4 - Prob. 17ECh. 6.4 - Prob. 20ECh. 6.4 - Prob. 23ECh. 6.4 - Prob. 24ECh. 6.5 - Let I=04f(x)dx, where f is the function whose...Ch. 6.5 - Prob. 2ECh. 6.5 - Estimate 01cos(x2)dx using (a) the Trapezoidal...Ch. 6.5 - Prob. 4ECh. 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. 11ECh. 6.5 - Prob. 12ECh. 6.5 - Use (a) the Midpoint Rule and (b) Simpsons Rule to...Ch. 6.5 - Prob. 14ECh. 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. 17ECh. 6.5 - Prob. 18ECh. 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. 30ECh. 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. 38ECh. 6.5 - Prob. 36ECh. 6.5 - Prob. 37ECh. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 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. 4ECh. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Determine whether each integral is convergent or...Ch. 6.6 - Prob. 7ECh. 6.6 - Prob. 8ECh. 6.6 - Prob. 9ECh. 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. 1RCCCh. 6 - How do you evaluate sinmxcosnxdx if m is odd? What...Ch. 6 - Prob. 3RCCCh. 6 - Prob. 4RCCCh. 6 - Prob. 5RCCCh. 6 - Prob. 6RCCCh. 6 - Prob. 7RCCCh. 6 - Prob. 8RCCCh. 6 - Prob. 1RQCh. 6 - Prob. 2RQCh. 6 - Prob. 3RQCh. 6 - Prob. 4RQCh. 6 - Prob. 5RQCh. 6 - Prob. 6RQCh. 6 - Prob. 7RQCh. 6 - Prob. 8RQCh. 6 - Prob. 9RQCh. 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)2xdxCh. 6 - Evaluate the integral. 2. 12x(x+1)2dxCh. 6 - Evaluate the integral. 0/2sinecosdCh. 6 - 41550-7-4RE
1–40 Evaluate the integral.
4.
Ch. 6 - Evaluate the integral. 5. dt2t2+3t+1Ch. 6 - Prob. 6RECh. 6 - Prob. 15RECh. 6 - Prob. 8RECh. 6 - Prob. 7RECh. 6 - Evaluate the integral. 10. 01arctanx1+x2dxCh. 6 - Prob. 11RECh. 6 - Prob. 12RECh. 6 - Prob. 13RECh. 6 - Prob. 14RECh. 6 - Evaluate the integral. 14x3/2lnxdxCh. 6 - Evaluate the integral. 16. sec6tan2dCh. 6 - Prob. 17RECh. 6 - Prob. 18RECh. 6 - Prob. 19RECh. 6 - Prob. 26RECh. 6 - Prob. 21RECh. 6 - Prob. 22RECh. 6 - Prob. 23RECh. 6 - Evaluate the integral. 24. excosxdxCh. 6 - Evaluate the integral. 25. 3x3x2+6x4(x2+1)(x2+2)dxCh. 6 - Prob. 20RECh. 6 - Prob. 27RECh. 6 - Prob. 28RECh. 6 - Prob. 29RECh. 6 - Prob. 30RECh. 6 - Prob. 31RECh. 6 - Prob. 32RECh. 6 - Prob. 33RECh. 6 - Prob. 34RECh. 6 - Prob. 35RECh. 6 - Prob. 36RECh. 6 - Prob. 37RECh. 6 - Prob. 38RECh. 6 - Prob. 39RECh. 6 - Prob. 40RECh. 6 - Prob. 41RECh. 6 - Prob. 42RECh. 6 - Evaluate the integral or show that it is...Ch. 6 - Prob. 44RECh. 6 - Prob. 45RECh. 6 - Prob. 46RECh. 6 - Prob. 47RECh. 6 - Prob. 48RECh. 6 - Prob. 49RECh. 6 - Prob. 50RECh. 6 - Prob. 51RECh. 6 - Prob. 52RECh. 6 - Prob. 53RECh. 6 - Prob. 54RECh. 6 - Prob. 55RECh. 6 - Prob. 56RECh. 6 - Prob. 57RECh. 6 - Prob. 58RECh. 6 - Prob. 59RECh. 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. 62RECh. 6 - Prob. 63RECh. 6 - Prob. 64RECh. 6 - Prob. 65RE
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