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Problem 5.5.36
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« sec 4w tan 4w dw
Problem 5.5.12
Use a substitution of the form u = aπ + b to evaluate the following indefinite integrals
β«(eΒ³Λ£ βΊΒΉ dπ
Problem 5.4.56
Average value of the derivative Suppose Ζ ' is a continuous function for all real numbers. Show that the average value of the derivative on an interval [a, b] is Ζβ»' = (Ζ(b) βΖ(a))/ (bβa) . Interpret this result in terms of secant lines.
Problem 5.2.35
Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ on [a,b]. Identify Ζ and express the limit as a definite integral.
n
lim β (πβ*Β² + 1) βπβ on [0,2]
β β 0 k=1
Problem 5.2.39
Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.
β«ββ΄ (8β2π) dπ
Problem 5.1.19
Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
v = [1 / (2t + 1)] (m/s), for 0 β€ t β€ 8 ; n = 4
Problem 5.1.61
{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.β
The right Riemann sum for Ζ(π)) = x + 1 on [0, 4] with n = 50.
Problem 5.3.5
The linear function Ζ(π) = 3 β π is decreasing on the interval [0, 3]. Is its area function for Ζ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.
Problem 5.5.43
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« sin π secβΈ π dπ
Problem 5.3.66
Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.
The region bounded by y = 6 cos π and the π-axis between π = βΟ/2 and π = Ο
Problem 5.5.80
Variations on the substitution method Evaluate the following integrals.
β« yΒ²/(y + 1)β΄ dy
Problem 5.5.106
General results Evaluate the following integrals in which the function Ζ is unspecified. Note that Ζβ½α΅βΎ is the pth derivative of Ζ and Ζα΅ is the pth power of Ζ. Assume Ζ and its derivatives are continuous for all real numbers.
β« (5 ΖΒ³ (π) + 7ΖΒ² (π) + Ζ (π )) Ζ'(π) dπ
Problem 5.4.33
Average distance on a parabola What is the average distance between the parabola y = 30π (20 β π ) and the π-axis on the interval [0, 20] ?
Problem 5.4.23
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«Β²ββ [(xΒ³ β 4x) / (xΒ² + 1)] dx
Problem 5.3.23
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure.
β«βΒΉ (πΒ² β 2π + 3) dπ
Problem 5.5.74
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β^Ο/β΄ eΛ’αΆ¦βΏΒ² Λ£ sin 2π dπ
Problem 5.5.40
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« (sinβ΅ π + 3 sinΒ³ πβ sin π) cos π dπ
Problem 5.3.84
Derivatives of integrals Simplify the following expressions.
d/dt β«βα΅ dπ/(1 + πΒ²) + β«βΒΉ/α΅ dx/(1 + πΒ²)
Problem 5.2.81
Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.
β«ββ· (4π + 6) dπ
Problem 5.3.31
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«ββΈ 8πΒΉ/Β³ dπ
Problem 5.4.49
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
β«α΅ββ Ζ(g(π)) dπ
Problem 5.5.46
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βΒΉ 2eΒ²Λ£ dπ
Problem 5.5.92
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« π cosΒ²πΒ² dπ
Problem 5.3.114
Max/min of area functions Suppose Ζ is continuous on [0 ,β) and A(π) is the net area of the region bounded by the graph of Ζ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of Ζ. Verify this fact with the function Ζ(π) = πΒ² - 10π.
Problem 5.3.103
{Use of Tech} Areas of regions Find the area of the region π bounded by the graph of Ζ and the π-axis on the given interval. Graph Ζ and show the region π .
Ζ(π) = 2 β |π| on [ β 2 , 4]
Problem 5.5.10
Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.
β« (6π + 1) β(3πΒ² + π) dπ , u = 3πΒ² + π
Problem 5.1.67
Identifying Riemann sums Fill in the blanks with an interval and a value of n.β
4
β Ζ (1.5 + k) β’ 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]
k = 1
with n = ________ .
Problem 5.5.70
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· dπ
Problem 5.3.8
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Problem 5.2.59
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βα΅ Ζ(π) dπ
