Single Variable Calculus
Single Variable Calculus
8th Edition
ISBN: 9781305266636
Author: James Stewart
Publisher: Cengage Learning
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Chapter 4.1, Problem 2E

(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12.

(i) L6 (sample points are left endpoints)

(ii) R6 (sample points are right endpoints)

(iii) M6 (sample points are midpoints)

(b) Is L6 an underestimate or overestimate of the true area?

(c) Is R6 an underestimate or overestimate of the true area?

(d) Which of the numbers L6, R6, or M6 gives the best estimate? Explain.

Chapter 4.1, Problem 2E, (a) Use six rectangles to find estimates of each type for the area under the given graph of f from x

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Chapter 4 Solutions

Single Variable Calculus

Ch. 4.1 - Prob. 1ECh. 4.1 - (a) Use six rectangles to find estimates of each...Ch. 4.1 - (a) Estimate the area under the graph of f(x) =...Ch. 4.1 - Prob. 4ECh. 4.1 - (a) Estimate the area under the graph of f(x) = 1...Ch. 4.1 - Prob. 6ECh. 4.1 - Prob. 7ECh. 4.1 - Prob. 8ECh. 4.1 - With a programmable calculator (or a computer), it...Ch. 4.1 - With a programmable calculator (or a computer), it...
Ch. 4.1 - Prob. 11ECh. 4.1 - Prob. 12ECh. 4.1 - The speed of a runner increased steadily during...Ch. 4.1 - Prob. 14ECh. 4.1 - Oil leaked from a tank at a rate of r(t) liters...Ch. 4.1 - Prob. 16ECh. 4.1 - The velocity graph of a braking car is shown. Use...Ch. 4.1 - The velocity graph of a car accelerating from rest...Ch. 4.1 - In someone infected with measles, the virus level...Ch. 4.1 - The table shows the number of people per day who...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Prob. 22ECh. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Prob. 24ECh. 4.1 - Prob. 25ECh. 4.1 - (a) Use Definition 2 to find an expression for the...Ch. 4.1 - Let A be the area under the graph of an increasing...Ch. 4.1 - If A is the area under the curve y = sin x from 0...Ch. 4.1 - (a) Express the area under the curve y = x5 from 0...Ch. 4.1 - (a) Express the area under the curve y = x4 + 5x2...Ch. 4.1 - Prob. 31ECh. 4.1 - (a) Let An be the area of a polygon with n equal...Ch. 4.2 - Evaluate the Riemann sum for f(x)=x1,6x4, with...Ch. 4.2 - If f(x)=cosx0x3/4 evaluate the Riemann sum with n...Ch. 4.2 - If f(x)=x24,0x3, find the Riemann sum with n = 6,...Ch. 4.2 - (a) Find the Riemann sum for f(x)=1/x,1x2, with...Ch. 4.2 - The graph of a function f is given. Estimate...Ch. 4.2 - The graph of g is shown. Estimate 24g(x)dx with...Ch. 4.2 - A table of values of an increasing function f is...Ch. 4.2 - Prob. 8ECh. 4.2 - Prob. 9ECh. 4.2 - Prob. 10ECh. 4.2 - Prob. 11ECh. 4.2 - Prob. 12ECh. 4.2 - If you have a CAS that evaluates midpoint...Ch. 4.2 - Prob. 14ECh. 4.2 - Prob. 15ECh. 4.2 - Prob. 16ECh. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Prob. 20ECh. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Prob. 26ECh. 4.2 - Prove that abxdx=b2a22.Ch. 4.2 - Prob. 28ECh. 4.2 - Prob. 29ECh. 4.2 - Prob. 30ECh. 4.2 - Express the integral as a limit of sums. Then...Ch. 4.2 - Prob. 32ECh. 4.2 - The graph of f is shown. Evaluate each integral by...Ch. 4.2 - The graph of g consists of two straight lines and...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Prob. 39ECh. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Prob. 41ECh. 4.2 - Give that 0sin4xdx=38, what is 0sin4d?Ch. 4.2 - In Example 4.1.2 we showed that 01x2dx=13. Use...Ch. 4.2 - Prob. 44ECh. 4.2 - Prob. 45ECh. 4.2 - Prob. 46ECh. 4.2 - Prob. 47ECh. 4.2 - If 28f(x)dx=7.3 and 24f(x)dx=5.9, find 48f(x)dx.Ch. 4.2 - Prob. 49ECh. 4.2 - Prob. 50ECh. 4.2 - Prob. 51ECh. 4.2 - Prob. 52ECh. 4.2 - Prob. 53ECh. 4.2 - Suppose f has absolute minimum value m and...Ch. 4.2 - Prob. 55ECh. 4.2 - Prob. 56ECh. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Prob. 58ECh. 4.2 - Prob. 59ECh. 4.2 - Prob. 60ECh. 4.2 - Prob. 61ECh. 4.2 - Prob. 62ECh. 4.2 - Prob. 63ECh. 4.2 - Prob. 64ECh. 4.2 - Prob. 65ECh. 4.2 - Prob. 66ECh. 4.2 - Prob. 67ECh. 4.2 - Prob. 68ECh. 4.2 - Prob. 69ECh. 4.2 - Prob. 70ECh. 4.2 - Prob. 71ECh. 4.2 - Prob. 72ECh. 4.2 - Express the limit as a definite integral. 73....Ch. 4.2 - Prob. 74ECh. 4.2 - Prob. 75ECh. 4.3 - Explain exactly what is meant by the statement...Ch. 4.3 - Prob. 2ECh. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Prob. 4ECh. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Prob. 6ECh. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 8ECh. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 10ECh. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 12ECh. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 16ECh. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 18ECh. 4.3 - Evaluate the integral. 19. 13(x2+2x4)dxCh. 4.3 - Prob. 20ECh. 4.3 - Prob. 21ECh. 4.3 - Prob. 22ECh. 4.3 - Evaluate the integral. 23. 19xdxCh. 4.3 - Prob. 24ECh. 4.3 - Evaluate the integral. 25. /6sindCh. 4.3 - Prob. 26ECh. 4.3 - Evaluate the integral. 27. 01(u+2)(u3)duCh. 4.3 - Prob. 28ECh. 4.3 - Evaluate the integral. 29. 142+x2xdxCh. 4.3 - Prob. 30ECh. 4.3 - Evaluate the integral. 31. /6/2csctcottdtCh. 4.3 - Prob. 32ECh. 4.3 - Prob. 33ECh. 4.3 - Prob. 34ECh. 4.3 - Evaluate the integral. 35. 12v5+3v6v4dvCh. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.3 - Prob. 38ECh. 4.3 - Prob. 39ECh. 4.3 - Prob. 40ECh. 4.3 - Prob. 41ECh. 4.3 - Prob. 42ECh. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Prob. 44ECh. 4.3 - Prob. 45ECh. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Prob. 47ECh. 4.3 - Prob. 48ECh. 4.3 - Prob. 49ECh. 4.3 - Prob. 50ECh. 4.3 - Prob. 51ECh. 4.3 - Prob. 52ECh. 4.3 - Prob. 53ECh. 4.3 - Prob. 54ECh. 4.3 - Prob. 55ECh. 4.3 - Prob. 56ECh. 4.3 - Prob. 57ECh. 4.3 - If f(x)=0x(1t2)cos2tdt, on what interval is f...Ch. 4.3 - On what interval is the curve y=0xt2t2+t+2dt...Ch. 4.3 - Prob. 60ECh. 4.3 - Prob. 61ECh. 4.3 - Prob. 62ECh. 4.3 - The Fresnel function S was defined in Example 3...Ch. 4.3 - The sine integral function Si(x)=0xsinttdt is...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Prob. 67ECh. 4.3 - Prob. 68ECh. 4.3 - Prob. 69ECh. 4.3 - Prob. 70ECh. 4.3 - Prob. 71ECh. 4.3 - (a) Show that cos(x2) cos x for 0 x 1. (b)...Ch. 4.3 - Show that 0510x2x4+x2+1dx0.1 by comparing the...Ch. 4.3 - Let f(x)={0ifx0xif0x12xif1x20ifx2 and...Ch. 4.3 - Find a function f and a number a such that...Ch. 4.3 - Prob. 76ECh. 4.3 - A manufacturing company owns a major piece of...Ch. 4.3 - A high-tech company purchases a new computing...Ch. 4.3 - Evaluate the integral. 79. 1912xdxCh. 4.3 - Prob. 80ECh. 4.3 - Prob. 81ECh. 4.3 - Prob. 82ECh. 4.3 - Prob. 83ECh. 4.3 - Prob. 84ECh. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Prob. 3ECh. 4.4 - Prob. 4ECh. 4.4 - Find the general indefinite integral. 5....Ch. 4.4 - Prob. 6ECh. 4.4 - Find the general indefinite integral. 7....Ch. 4.4 - Prob. 8ECh. 4.4 - Find the general indefinite integral. 9....Ch. 4.4 - Prob. 10ECh. 4.4 - Find the general indefinite integral. 11. 1+x+xxdxCh. 4.4 - Prob. 12ECh. 4.4 - Find the general indefinite integral. 13....Ch. 4.4 - Prob. 14ECh. 4.4 - Find the general indefinite integral. 15....Ch. 4.4 - Prob. 16ECh. 4.4 - Find the general indefinite integral. Illustrate...Ch. 4.4 - Prob. 18ECh. 4.4 - Evaluate the integral. 19. 23(x23)dxCh. 4.4 - Prob. 20ECh. 4.4 - Evaluate the integral. 21. 20(12t4+14t3t)dtCh. 4.4 - Prob. 22ECh. 4.4 - Evaluate the integral. 23. 02(2x3)(4x2+1)dxCh. 4.4 - Prob. 24ECh. 4.4 - Evaluate the integral. 25. 0(4sin3cos)dCh. 4.4 - Prob. 26ECh. 4.4 - Prob. 27ECh. 4.4 - Prob. 28ECh. 4.4 - Prob. 29ECh. 4.4 - Prob. 30ECh. 4.4 - Prob. 31ECh. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - Prob. 34ECh. 4.4 - Prob. 35ECh. 4.4 - Prob. 36ECh. 4.4 - Prob. 37ECh. 4.4 - Prob. 38ECh. 4.4 - Prob. 39ECh. 4.4 - Prob. 40ECh. 4.4 - Prob. 41ECh. 4.4 - Prob. 42ECh. 4.4 - Prob. 43ECh. 4.4 - Repeat Exercise 43 for the curve y = 2x + 3x4 ...Ch. 4.4 - The area of the region that lies to the right of...Ch. 4.4 - Prob. 46ECh. 4.4 - If w'(t) is the rate of growth of a child in...Ch. 4.4 - Prob. 48ECh. 4.4 - If oil leaks from a tank at a rate of r(t) gallons...Ch. 4.4 - A honeybee population starts with 100 bees and...Ch. 4.4 - In Section 3.7 we defined the marginal revenue...Ch. 4.4 - Prob. 52ECh. 4.4 - If x is measured in meters and f(x) is measured in...Ch. 4.4 - If the units for x are feet and the units for a(x)...Ch. 4.4 - The velocity function (in meters per second) is...Ch. 4.4 - The velocity function (in meters per second) is...Ch. 4.4 - Prob. 57ECh. 4.4 - Prob. 58ECh. 4.4 - The linear density of a rod of length 4 m is given...Ch. 4.4 - Water flows from the bottom of a storage tank at a...Ch. 4.4 - Prob. 61ECh. 4.4 - Suppose that a volcano is erupting and readings of...Ch. 4.4 - Prob. 63ECh. 4.4 - Prob. 64ECh. 4.4 - The graph of the acceleration a(t) of a car...Ch. 4.4 - Shown is the graph of traffic on an Internet...Ch. 4.4 - The following graph shows the power consumption in...Ch. 4.4 - Prob. 68ECh. 4.4 - Evaluate the integral. 69. (sinx+sinhx)dxCh. 4.4 - Prob. 70ECh. 4.4 - Prob. 71ECh. 4.4 - Prob. 72ECh. 4.4 - Prob. 73ECh. 4.4 - The area labeled B is three times the area labeled...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 2ECh. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 4ECh. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 6ECh. 4.5 - Evaluate the indefinite integral. 7. x1x2dxCh. 4.5 - Prob. 8ECh. 4.5 - Evaluate the indefinite integral. 9. (12x)9dxCh. 4.5 - Prob. 10ECh. 4.5 - Evaluate the indefinite integral. 11. sin(2/3)dCh. 4.5 - Prob. 12ECh. 4.5 - Evaluate the indefinite integral. 13. sec3ttan3tdtCh. 4.5 - Prob. 14ECh. 4.5 - Evaluate the indefinite integral. 15. cos(1+5t)dtCh. 4.5 - Prob. 16ECh. 4.5 - Evaluate the indefinite integral. 17. sec2tan3dCh. 4.5 - Prob. 18ECh. 4.5 - Evaluate the indefinite integral. 19....Ch. 4.5 - Prob. 20ECh. 4.5 - Prob. 21ECh. 4.5 - Evaluate the indefinite integral. 22. cos(/x)x2dxCh. 4.5 - Evaluate the indefinite integral. 23. z21+z33dzCh. 4.5 - Prob. 24ECh. 4.5 - Evaluate the indefinite integral. 25. cotxcsc2xdxCh. 4.5 - Prob. 26ECh. 4.5 - Evaluate the indefinite integral. 27. sec3xtanxdxCh. 4.5 - Prob. 28ECh. 4.5 - Evaluate the indefinite integral. 29. x(2x+5)8dxCh. 4.5 - Prob. 30ECh. 4.5 - Prob. 31ECh. 4.5 - Prob. 32ECh. 4.5 - Prob. 33ECh. 4.5 - Prob. 34ECh. 4.5 - Evaluate the definite integral. 35. 01cos(t/2)dtCh. 4.5 - Prob. 36ECh. 4.5 - Evaluate the definite integral. 37. 011+7x3dxCh. 4.5 - Prob. 38ECh. 4.5 - Prob. 39ECh. 4.5 - Prob. 40ECh. 4.5 - Prob. 41ECh. 4.5 - Prob. 42ECh. 4.5 - Evaluate the definite integral. 43. 013dx(1+2x)23Ch. 4.5 - Prob. 44ECh. 4.5 - Evaluate the definite integral. 45. 0axx2+a2dx(a0)Ch. 4.5 - Prob. 46ECh. 4.5 - Evaluate the definite integral. 47. 12xx1dxCh. 4.5 - Prob. 48ECh. 4.5 - Evaluate the definite integral. 49....Ch. 4.5 - Prob. 50ECh. 4.5 - Prob. 51ECh. 4.5 - Prob. 52ECh. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Prob. 55ECh. 4.5 - Prob. 56ECh. 4.5 - Breathing is cyclic and a full respiratory cycle...Ch. 4.5 - A model for the basal metabolism rate, in kcal/h,...Ch. 4.5 - Prob. 59ECh. 4.5 - Prob. 60ECh. 4.5 - Prob. 61ECh. 4.5 - Prob. 62ECh. 4.5 - If a and b are positive numbers, show that...Ch. 4.5 - If f is continuous on [0, ], use the substitution...Ch. 4.5 - If f is continuous, prove that...Ch. 4.5 - Prob. 66ECh. 4.5 - Prob. 67ECh. 4.5 - Prob. 68ECh. 4.5 - Prob. 69ECh. 4.5 - Prob. 70ECh. 4.5 - Prob. 71ECh. 4.5 - Prob. 72ECh. 4.5 - Prob. 73ECh. 4.5 - Prob. 74ECh. 4.5 - Prob. 75ECh. 4.5 - Evaluate the integral. 76. sin(lnx)xdxCh. 4.5 - Prob. 77ECh. 4.5 - Prob. 78ECh. 4.5 - Prob. 79ECh. 4.5 - Prob. 80ECh. 4.5 - Prob. 81ECh. 4.5 - Prob. 82ECh. 4.5 - Evaluate the integral. 83. 01ez+1ez+zdzCh. 4.5 - Prob. 84ECh. 4.5 - Prob. 85ECh. 4 - (a) Write an expression for a Riemann sum of a...Ch. 4 - (a) Write the definition of the definite integral...Ch. 4 - State the Midpoint Rule.Ch. 4 - State both parts of the Fundamental Theorem of...Ch. 4 - (a) State the Net Change Theorem. (b) If r(t) is...Ch. 4 - Suppose a particle moves back and forth along a...Ch. 4 - Prob. 7RCCCh. 4 - Explain exactly what is meant by the statement...Ch. 4 - Prob. 9RCCCh. 4 - Prob. 1RQCh. 4 - Prob. 2RQCh. 4 - Prob. 3RQCh. 4 - Prob. 4RQCh. 4 - Prob. 5RQCh. 4 - Prob. 6RQCh. 4 - Prob. 7RQCh. 4 - Prob. 8RQCh. 4 - Prob. 9RQCh. 4 - Determine whether the statement is true or false....Ch. 4 - Prob. 11RQCh. 4 - Prob. 12RQCh. 4 - Prob. 13RQCh. 4 - Prob. 14RQCh. 4 - Prob. 15RQCh. 4 - Prob. 16RQCh. 4 - Prob. 17RQCh. 4 - Determine whether the statement is true or false....Ch. 4 - Use the given graph of f to find the Riemann sum...Ch. 4 - Prob. 2RECh. 4 - Prob. 3RECh. 4 - Prob. 4RECh. 4 - Prob. 5RECh. 4 - Prob. 6RECh. 4 - The figure shows the graphs of f,f, and 0xf(t)dt....Ch. 4 - Evaluate: (a) 0/2ddx(sinx2cosx3)dx (b)...Ch. 4 - Prob. 9RECh. 4 - Prob. 10RECh. 4 - Prob. 11RECh. 4 - Prob. 12RECh. 4 - Prob. 13RECh. 4 - Prob. 14RECh. 4 - Prob. 15RECh. 4 - Prob. 16RECh. 4 - Prob. 17RECh. 4 - Prob. 18RECh. 4 - Prob. 19RECh. 4 - Prob. 20RECh. 4 - Prob. 21RECh. 4 - Prob. 22RECh. 4 - Prob. 23RECh. 4 - Prob. 24RECh. 4 - Prob. 25RECh. 4 - Prob. 26RECh. 4 - Prob. 27RECh. 4 - Prob. 28RECh. 4 - Prob. 29RECh. 4 - Prob. 30RECh. 4 - Prob. 31RECh. 4 - Prob. 32RECh. 4 - Prob. 33RECh. 4 - Prob. 34RECh. 4 - Prob. 35RECh. 4 - Prob. 36RECh. 4 - Prob. 37RECh. 4 - Prob. 38RECh. 4 - Prob. 39RECh. 4 - Prob. 40RECh. 4 - Prob. 41RECh. 4 - Prob. 42RECh. 4 - Prob. 43RECh. 4 - Prob. 44RECh. 4 - Prob. 45RECh. 4 - A particle moves along a line with velocity...Ch. 4 - Prob. 47RECh. 4 - Prob. 48RECh. 4 - A population of honeybees increased at a rate of...Ch. 4 - Prob. 50RECh. 4 - If f is continuous and 02f(x)dx=6, evaluate...Ch. 4 - The Fresnel function S(x)=0xsin(12t2)dt was...Ch. 4 - Prob. 53RECh. 4 - Prob. 54RECh. 4 - Prob. 55RECh. 4 - Prob. 56RECh. 4 - If f is continuous on [0, 1], prove that...Ch. 4 - Prob. 58RECh. 4 - Prob. 1PCh. 4 - Prob. 2PCh. 4 - Prob. 3PCh. 4 - (a) Graph several members of the family of...Ch. 4 - Prob. 5PCh. 4 - If f(x)=0xx2sin(t2)dt, find f(x).Ch. 4 - Prob. 7PCh. 4 - Prob. 8PCh. 4 - Prob. 9PCh. 4 - Find d2dx20x(1sint1+u4du)dt.Ch. 4 - Suppose the coefficients of the cubic polynomial...Ch. 4 - Prob. 12PCh. 4 - Prob. 13PCh. 4 - The figure shows a parabolic segment, that is. a...Ch. 4 - Prob. 15PCh. 4 - Prob. 16PCh. 4 - Evaluate limn(1nn+1+1nn+2++1nn+n).Ch. 4 - For any number c, we let fc(x) be the smaller of...

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