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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
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 5.1.61
{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 5.5.74
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 5.2.39
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 5.3.5
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 5.2.35
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 5.5.40
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 5.5.43
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 5.1.19
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 5.4.33
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 5.5.36
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 5.3.23
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 5.4.23
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«Β²ββ [(xΒ³ β 4x) / (xΒ² + 1)] dx
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 5.5.12
Use a substitution of the form u = aπ + b to evaluate the following indefinite integrals
β«(eΒ³Λ£ βΊΒΉ dπ
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 5.3.114
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 5.5.46
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βΒΉ 2eΒ²Λ£ dπ
Problem 5.5.70
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· dπ
Problem 5.3.103
{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 5.3.8
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Problem 5.3.43
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«βββ»ΒΉ πβ»Β³ dπ
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 5.4.49
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 5.5.92
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« π cosΒ²πΒ² dπ
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
