Calculus Volume 22nd EditionGilbert Strang, Edwin Jed Herman Publisher: OpenStax College.ISBN: 9781630182021
Calculus Volume 2
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
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
17th Edition
ISBN: 9781938168062
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