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Problem 5.3.51
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«ββ΄ (π β 2)/βπ 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 5.5.18
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« πeΛ£Β² dπ
Problem 5.5.80
Variations on the substitution method Evaluate the following integrals.
β« yΒ²/(y + 1)β΄ dy
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 5.2.67
Use geometry and properties of integrals to evaluate
β«βΒΉ (2π + β(1βπΒ²) + 1) dπ
Problem 5.5.29
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« (πβΆ β 3πΒ²)β΄ (πβ΅ β π) dπ
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 5.5.20
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 5.5.36
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 5.4.31
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) = πβΏ on [0,1] , for any positive integer n
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 5.2.87
Area by geometry Use geometry to evaluate the following integrals.
β«β΄ββ β(24 β 2π β πΒ²) dπ
Problem 5.2.81
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 5.5.81
Variations on the substitution method Evaluate the following integrals.
β« π/(βπ + 4) dπ
Problem 5.5.3
The composite function Ζ(g(π)) consists of an inner function g and an outer function Ζ. If an integrand includes Ζ(g(π)), which function is often a likely choice for a new variable u?
Problem 5.3.11
Evaluate β«ββΈ Ζ β²(t) dt , where Ζ β² is continuous on [3, 8], Ζ(3) = 4, and Ζ(8) = 20 .
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 5.2.83
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 5.1.61
{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 5.3.35
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«ββΉ 2/(βπ) dπ
Problem 5.2.37
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 π*β (ln π*β) βπβ on [1,2]
β β 0 k=1
Problem 5.5.11
Use a substitution of the form u = aπ + b to evaluate the following indefinite integrals.
β«(π + 1)ΒΉΒ² 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 5.5.1
On which derivative rule is the Substitution Rule based?
Problem 5.5.66
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βα΅Β² (ln p)/p dp
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 5.5.96
Areas of regions Find the area of the following regions.
The region bounded by the graph of Ζ(π) = x /β(πΒ² β9) and the π-axis between and π = 4 and π= 5
Problem 5.3.55
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
β«Ο/β^Β³Ο/β΄ (cotΒ² π + 1) dπ
Problem 5.5.84
Variations on the substitution method Evaluate the following integrals.
β« (π΅ + 1) β(3π΅ + 2) dπ΅
