Solutions for Calculus Volume 2
Problem 60E:
In the following exercises, express the limits as integrals. 60. limni=1n(xi*)x over [1, 3]Problem 61E:
In the following exercises, express the limits as integrals. 61. limni=1n(5( x i * )23( x i * )3)x...Problem 62E:
In the following exercises, express the limits as integrals. 62. limni=1nsin2x(2xi*)x over [0, 1]Problem 63E:
In the following exercises, express the limits as integrals. 63. limni=1ncos2x(2xi*)x over [0, 1]Problem 64E:
In the following exercises, given Ln or Rn as indicated, express their limits as n as definite...Problem 65E:
In the following exercises, given Ln or Rn as indicated, express their limits as n as definite...Problem 66E:
In the following exercises, given Ln or Rn as indicated, express their limits as n as definite...Problem 67E:
In the following exercises, given Ln or Rn as indicated, express their limits as n as definite...Problem 68E:
In the following exercises, given Ln or Rn as indicated, express their limits as n as definite...Problem 69E:
In the following exercises, given Ln or Rn as indicated, express their limits as n as definite...Problem 70E:
In the following exercises, evaluate the integrals of the functions graphed using the formulas for...Problem 71E:
In the following exercises, evaluate the integrals of the functions graphed using the formulas for...Problem 72E:
In the following exercises, evaluate the integrals of the functions graphed using the formulas for...Problem 73E:
In the following exercises, evaluate the integrals of the functions graphed using the formulas for...Problem 74E:
In the following exercises, evaluate the integrals of the functions graphed using the formulas for...Problem 75E:
In the following exercises, evaluate the integrals of the functions graphed using the formulas for...Problem 81E:
In the following exercises, evaluate the integral using area formulas. 81. 154 ( x3 )2dxProblem 82E:
In the following exercises, evaluate the integral using area formulas. 82. 01236 ( x6 )2dxProblem 84E:
In the following exercises, use averages of values at the left (L) and tight (R) endpoints to...Problem 85E:
In the following exercises, use averages of values at the left (L) and tight (R) endpoints to...Problem 86E:
In the following exercises, use averages of values at the left (L) and tight (R) endpoints to...Problem 87E:
In the following exercises, use averages of values at the left (L) and tight (R) endpoints to...Problem 88E:
Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,...Problem 89E:
Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,...Problem 90E:
Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,...Problem 91E:
Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,...Problem 92E:
Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,...Problem 93E:
Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,...Problem 94E:
In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals....Problem 95E:
In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals....Problem 96E:
In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals....Problem 97E:
In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals....Problem 98E:
In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals. 98....Problem 99E:
In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals. 99....Problem 100E:
In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals....Problem 101E:
In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals....Problem 102E:
In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals....Problem 103E:
In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals....Problem 104E:
In the following exercises, use the comparison theorem. 104. Show that 03(x26x+9)dx0 .Problem 105E:
In the following exercises, use the comparison theorem. 105. Show that 23(x3)(x+2)dx0 .Problem 106E:
In the following exercises, use the comparison theorem. 106. Show that 011+x3dx011+x2dx .Problem 107E:
In the following exercises, use the comparison theorem. 107. Show that 121+xdx121+x2dx .Problem 108E:
In the following exercises, use the comparison theorem. 108. Show that 0/2sintdt4 . (Hint: sint2t...Problem 109E:
In the following exercises, use the comparison theorem. 109. Show that /4/4costdt2/4 .Problem 110E:
In the following exercises, find 1112 average value fave of f between a and b, and find a point c,...Problem 111E:
In the following exercises, find 1112 average value fave of f between a and b, and find a point c,...Problem 112E:
In the following exercises, find 1112 average value fave of f between a and b, and find a point c,...Problem 113E:
In the following exercises, find 1112 average value fave of f between a and b, and find a point c,...Problem 114E:
In the following exercises, find 1112 average value fave of f between a and b, and find a point c,...Problem 115E:
In the following exercises, find 1112 average value fave of f between a and b, and find a point c,...Problem 116E:
In the following exercises, approximate the average value using Riemann sums L100 and R100. How does...Problem 117E:
In the following exercises, approximate the average value using Riemann sums L100 and R100. How does...Problem 118E:
In the following exercises, approximate the average value using Riemann sums L100 and R100. How does...Problem 119E:
In the following exercises, approximate the average value using Riemann sums L100 and R100. How does...Problem 120E:
In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,...Problem 121E:
In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,...Problem 122E:
In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,...Problem 123E:
In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,...Problem 124E:
In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,...Problem 125E:
In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,...Problem 126E:
Show that the average value of sin2t over [0, 2 ] is equal to 1/2 Without further calculation,...Problem 127E:
Show that the average value of cos2t over [0, 2 ] is equal to 1/2. Without further calculation,...Problem 128E:
Explain why the graphs of a quadratic function (parabola) p(x) and a linear function l (x) can...Problem 129E:
Suppose that parabola p(x)=ax2+bx+c opens downward (a < 0) and has a vertex of y=b2a0 . For which...Problem 130E:
Suppose [a, b} can be subdivided into subintervals a=a0a1a2...aN=b such that either f0 over [ai1,ai]...Problem 131E:
Suppose f and g are continuous functions such that cdf(t)dtcdg(t)dt for every subinterval [c, d] of...Problem 132E:
Suppose the average value of f over [a, b] is 1 and the average value of f over [b, c] is 1 where a...Problem 133E:
Suppose that [11. b] can be partitioned, taking a=a0a1...aN=b such that the average value of f over...Problem 134E:
Suppose that for each i such that 1iN one has i1if(t)dt=i . Show that 0Nf(t)dt=N( N+1)2 .Problem 135E:
Suppose that for each i such that 1iN one has i1if(t)dt=i2 . Show that 0Nf(t)dt=N( N+1)( 2N+1)6 .Problem 136E:
[T] Compute the left and right Riemann sums L10 and R10, and their average L10+R102 for f(t)=t2 over...Problem 137E:
[T] Compute the left and right Riemann sums, L10 and R10, and their average L10+R102 for f(t)=(4t2)...Problem 138E:
If 151+t4dt=41.7133... , what is 151+u4du ?Problem 139E:
Estimate 01tdt using the left and light endpoint sums, each with a single rectangle. How does the...Problem 140E:
Estimate 01tdt by comparison with the area of a single rectangle with height equal to the value of t...Problem 141E:
From the graph of sin(2(x) shown: a. Explain why 01sin(2t)dt=0 . b. Explain why, in general,...Browse All Chapters of This Textbook
Chapter 1 - IntegrationChapter 1.1 - Approximating AreasChapter 1.2 - The Definite IntegralChapter 1.3 - The Fundamental Theorem Of CalculusChapter 1.4 - Integration Formulas And The Net Change TheoremChapter 1.5 - SubstitutionChapter 1.6 - Integrals Involving Exponential And Logarithmic FunctionsChapter 1.7 - Integrals Resulting In Inverse Trigonometric FunctionsChapter 2 - Applications Of IntegrationChapter 2.1 - Areas Between Curves
Chapter 2.2 - Determining Volumes By SlicingChapter 2.3 - Volumes Of Revolution: Cylindrical ShellsChapter 2.4 - Am Length Of A Curve And Surface AreaChapter 2.5 - Physical ApplicationsChapter 2.6 - Moments And Centers Of MassChapter 2.7 - Integrals, Exponential Functions, And LogarithmsChapter 2.8 - Exponential Growth And DecayChapter 2.9 - Calculus Of The Hyperbolic FunctionsChapter 3 - Techniques Of IntegrationChapter 3.1 - Integration By PartsChapter 3.2 - Trigonometric IntegralsChapter 3.3 - Trigonometric SubstitutionChapter 3.4 - Partial FractionsChapter 3.5 - Other Strategies For IntegrationChapter 3.6 - Numerical IntegrationChapter 3.7 - Improper IntegralsChapter 4 - Introduction To Differential EquationsChapter 4.1 - Basics Of Differential EquationsChapter 4.2 - Direction Fields And Numerical MethodsChapter 4.3 - Separable EquationsChapter 4.4 - The Logistic EquationChapter 4.5 - First-order Linear EquationsChapter 5 - Sequences And SeriesChapter 5.1 - SequencesChapter 5.2 - Infinite SeriesChapter 5.3 - The Divergence And Integral TestsChapter 5.4 - Comparison TestsChapter 5.5 - Alternating SeriesChapter 5.6 - Ratio And Root TestsChapter 6 - Power SeriesChapter 6.1 - Power Series And FunctionsChapter 6.2 - Properties Of Power SeriesChapter 6.3 - Taylor And Maclaurin SeriesChapter 6.4 - Working With Taylor SeriesChapter 7 - Parametric Equations And Polar CoordinatesChapter 7.1 - Parametric EquationsChapter 7.2 - Calculus Of Parametric CurvesChapter 7.3 - Polar CoordinatesChapter 7.4 - Area And Arc Length In Polar CoordinatesChapter 7.5 - Conic Sections
Book Details
Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 2 covers integration, differential equations, sequences and series, and parametric equations and polar coordinates.
Sample Solutions for this Textbook
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More Editions of This Book
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Calculus Volume 2 by OpenStax
17th Edition
ISBN: 9781506698076
Calculus Volume 2
2nd Edition
ISBN: 9781630182021
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