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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 5.5.63
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β/β^ΒΉ/βΒ³ 4/(9πΒ² + 1) 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 5.2.79
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 5.5.20
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 5.3.23
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 5.4.31
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 5.4.23
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«Β²ββ [(xΒ³ β 4x) / (xΒ² + 1)] dx
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 5.5.57
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 5.4.39
Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.
Ζ(π) = 8 β 2π on [0, 4]
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 5.5.81
Variations on the substitution method Evaluate the following integrals.
β« π/(βπ + 4) dπ
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 5.4.47
Gateway Arch The Gateway Arch in St. Louis is 630 ft high and has a 630-ft base. Its shape can be modeled by the parabola y = 630 (1β (π/315)Β²) . Find the average height of the arch above the ground.
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 5.3.11
Evaluate β«ββΈ Ζ β²(t) dt , where Ζ β² is continuous on [3, 8], Ζ(3) = 4, and Ζ(8) = 20 .
Problem 5.2.37
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 5.4.38
Average height of a wave The surface of a water wave is described by y = 5 (1 + cos π) , for β Ο β€ π β€ Ο, where y = 0 corresponds to a trough of the wave (see figure). Find the average height of the wave above the trough on [ βΟ , Ο] .
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 5.5.23
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 5.1.1
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 5.5.17
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« 2π(πΒ² β 1)βΉβΉ dπ
Problem 5.5.79
Variations on the substitution method Evaluate the following integrals.
β« π/(βπβ4) dπ
Problem 5.5.84
Variations on the substitution method Evaluate the following integrals.
β« (π΅ + 1) β(3π΅ + 2) dπ΅
