Back
Problem 5.2.77a
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
β«ββ΄ (4πβ πΒ²) dπ
Problem 5.2.25b
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
Ζ(π) = tanβ»ΒΉ (3x - 1) on [0,1]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
Problem 5.2.51a
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(a) β«ββ° 3π(4 β π) d(π)
Problem 5.2.51b
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(b) β«ββ΄ π(π β 4) d(π)
Problem 5.2.53d
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(a) β«βΒ³ 5Ζ(π) dπ
Problem 5.2.53b
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(b) β«ββΆ (β3g(π)) dπ
Problem 5.2.53c
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(c) β«ββΆ (3Ζ(π) β g(π)) dπ
Problem 5.2.53.d
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(d) β«βΒ³ (Ζ(π) + 2g(π)) dπ
Problem 5.2.55a
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(a) β«ββ΄ 3f(π) dπ
Problem 5.2.55b
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(b) β«ββΆ (f(π) β g(π)) dπ
Problem 5.2.55d
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(d) β«ββΆ (g(π) β f(π) dπ
Problem 5.2.55f
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(f) β«βΒΉ 2f(π) dπ
Problem 5.2.57a
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«βΒΉ (πΒ³ β 2π) dπ = β3/4
(a) β«βΒΉ (4πβ2πΒ³) dπ
Problem 5.2.57b
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«βΒΉ (πΒ³ β 2π) dπ = β3/4
(b) β«ββ° (2πβπΒ³) dπ
Problem 5.2.58a
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«β^Ο/2 (cos ΞΈ β 2 sin ΞΈ) dΞΈ = β1
(a) β«β^Ο/2 (2 sin ΞΈ β cos ΞΈ) dΞΈ
Problem 5.2.58b
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«β^Ο/2 (cos ΞΈ β 2 sin ΞΈ) dΞΈ = β1
(b) β«β^Ο/2 (4 cos ΞΈ β 8 sin ΞΈ) dΞΈ
Problem 5.2.31c
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
β«ββΆ (1β2π) dπ ; n = 6
Problem 5.2.31d
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.
β«ββΆ (1β2π) dπ ; n = 6
Problem 5.2.32c
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
β«βΒ² (πΒ²β2) dπ ; n = 4
Problem 5.2.32d
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..
β«βΒ² (πΒ²β2) dπ ; n = 4
Problem 5.2.69d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(d) If β«βα΅ Ζ(π) dπ = β«βα΅ Ζ(π) dπ, then Ζ is a constant function.
Problem 5.2.33c
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
β«ββ· 1/π dπ ; n = 6
Problem 5.2.33d
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.
β«ββ· 1/π dπ ; n = 6
Problem 5.2.34c
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
β«β^Ο/2 cos π dπ ; n = 4
Problem 5.2.34d
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.
β«β^Ο/2 cos π dπ ; n = 4
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 5.2.83
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 5.2.81
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 5.2.79
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 5.2.21b
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
Ζ(x) = 4 - 2x on [0,4]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.