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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 5.5.20
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 5.4.18
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«βΟ/β^Ο/Β² 5 sin ΞΈ dΞΈ
Problem 5.3.2
Suppose F is an antiderivative of Ζ and A is an area function of Ζ. What is the relationship between F and A?
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 5.5.32
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 5.5.116
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 5.5.70
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· dπ
Problem 5.5.1
On which derivative rule is the Substitution Rule based?
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 5.3.106
{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) = πΒ² (π β 2) on [ β1 , 3]
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 5.2.61
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«βαΆ Ζ(π) dπ
Problem 5.5.84
Variations on the substitution method Evaluate the following integrals.
β« (π΅ + 1) β(3π΅ + 2) dπ΅
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 5.5.59
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β/ββ βββ^Β²/β΅ dπ/ xβ(25πΒ²β 1)
Problem 5.3.8
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
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 5.5.35
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« π csc πΒ² cot 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 5.3.10
Explain why β«βα΅ Ζ β²(π) dπ = Ζ(b) β Ζ(a)
Problem 5.1.75
Displacement from velocity The following functions describe the velocity of a car (in mi/hr) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval [0,t], where 0 β€ t β€ 3.
v(t) = { 30 if 0 β€ t β€ 2
50 if 2 < t < 2.5
44 if 2.5 < t β€ 3
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 5.1.23
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 5.3.100
Determine the intervals on which the function g(π) = β«ββ° t / (tΒ² + 1) dt is concave up or concave down.
Problem 5.5.87
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β«βΟ^Ο cosΒ² π dπ
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 5.5.22
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 5.2.67
Use geometry and properties of integrals to evaluate
β«βΒΉ (2π + β(1βπΒ²) + 1) dπ
