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Problem 5.5.20
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« [(βπ + 1)β΄ / 2βπ 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 5.1.33
A midpoint Riemann sum Approximate the area of the region bounded by the graph of Ζ(π) = 100 β xΒ² and the x-axis on [0, 10] with n = 5 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure).
Problem 5.5.57
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«Ο/β^Ο/Β² (cos π) / (sinΒ² π) dπ
Problem 5.5.23
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« πΒ³ (πβ΄ + 16)βΆ 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 5.4.18
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«βΟ/β^Ο/Β² 5 sin ΞΈ dΞΈ
Problem 5.5.32
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« 2 / (πβ4πΒ² β1) dπ , π > Β½
Problem 5.4.25
Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
Ζ(π) = πΒ³ on [β1, 1]
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 5.4.21
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«βΟ/β^Ο/β΄ secΒ² x dx
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 5.4.35
Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = tΒ² + 3t. Find the average velocity of the object over this time interval.
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 5.5.117
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
β« dπ / [β1 + β(1 + π)] (Hint: Begin with u = β(1 + π .)
Problem 5.2.83
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΄ (πΒ²β1) dπ
Problem 5.3.8
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Problem 5.3.10
Explain why β«βα΅ Ζ β²(π) dπ = Ζ(b) β Ζ(a)
Problem 5.2.87
Area by geometry Use geometry to evaluate the following integrals.
β«β΄ββ β(24 β 2π β πΒ²) dπ
Problem 5.2.79
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«βΒ² (2π + 1) dπ
Problem 5.3.93
Area functions from graphs The graph of Ζ is given in the figure. A(π) = β«βΛ£ Ζ(t) dt and evaluate A(2), A(5), A(8), and A(12).ββ
Problem 5.5.22
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« [ 1/(10πβ3) dπ
Problem 5.5.87
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β«βΟ^Ο cosΒ² π dπ
Problem 5.1.23
Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.
Ζ(π) = x + 1 on [1,6] ; n = 5
Problem 5.4.15
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«Β²ββ (xΒ² + xΒ³) dx
Problem 5.5.116
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
β« π sinβ΄ πΒ² cos πΒ² dπ (Hint: Begin with u = πΒ², and then use v = sin u .)
Problem 5.5.1
On which derivative rule is the Substitution Rule based?
Problem 5.5.37
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« secΒ² (10π + 7) dπ
Problem 5.1.1
Suppose an object moves along a line at 15 m/s, for 0 β€ t < 2 and at 25 m/s, for 2 β€ t β€ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 β€ t β€ 5.
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
