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Problem 5.1.47b
Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(b) 4 + 5 + 6 + 7 + 8 + 9
Problem 5.1.59b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].
Problem 5.1.53b
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
Ζ(π) = xΒ² β 1 on [2,5] ; n = 75
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
Problem 5.2.71b
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
β«βΒΉ (πΒ² + 1) dπ
Problem 5.1.51b
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
Ζ(π) = 3 βx on [0,4] ; n = 40
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
Problem 5.3.95b
Working with area functions Consider the function Ζ and the points a, b, and c.
(b) Graph Ζ and A.
Ζ(π) = eΛ£ ; a = 0 , b = ln 2 , c = ln 4
Problem 5.1.71b
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(b) Use geometry to find the displacement of the object between t = 0 and t = 2.
Problem 5.3.17b
Area functions for the same linear function Let Ζ(t) = t and consider the two area functions A(π) = β«βΛ£ Ζ(t) dt and F(π) = β«βΛ£ Ζ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (π) , for π β₯ 2.
Problem 5.2.75b
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
β«ββ΄ 2βπ dπ
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 5.2.23b
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
f(x) = sin 2x on [0,3Ο/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.
Problem 5.1.49b
Sigma notation Evaluate the following expressions.
(b) 10
β (2ΞΊ + 1)
ΞΊ=1
Problem 5.3.102b
{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
(b) Calculate g'(π)
g(π) = β«βΛ£ sin (ΟtΒ² ) dt ( a Fresnel integral)
Problem 5.3.87b
Matching functions with area functions Match the functions Ζ, whose graphs are given in aβ d, with the area functions A (π) = β«βΛ£ Ζ(t) dt, whose graphs are given in AβD.
Problem 5.5.95b
Explain βwhy or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ, Ζ', and Ζ'' are continuous functions for all real numbers.
(b) β« (Ζ(π))βΏ Ζ'(π) dπ = 1/(n + 1) (Ζ(π))βΏβΊΒΉ + C , n β β1 .
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 5.3.22b
Area functions for linear functions Consider the following functions Ζ and real numbers a (see figure).
b) Verify that A'(π) = Ζ(π).
Ζ(t) = 4t + 2 , a = 0
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 5.2.21b
{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.
Problem 5.3.98b
Working with area functions Consider the function Ζ and the points a, b, and c.
(b) Graph Ζ and A.
Ζ(π) = 1/π ; a = 1 , b = 4 , c = 6
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 5.2.77b
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
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 5.2.73b
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
β«βΒΉ cos β»ΒΉ π dπ
Problem 5.2.69b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) If Ζ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of β«βα΅ Ζ(π) dπ, for any positive integer n.
Problem 5.5.100b
Substitutions Suppose Ζ is an even function with β«ββΈ Ζ(π) dπ = 9 . Evaluate each integral.
(b) β«Β²ββ πΒ²Ζ(πΒ³) dπ
Problem 5.4.3b
Suppose Ζ is an even function and β«βΈββ Ζ(π) dπ = 18
(b) Evaluate β«βββΈ πΖ(π) dπ .
Problem 5.3.88b
Working with area functions Consider the function Ζ and its graph.
(b) Estimate the points (if any) at which A has a local maximum or minimum.
Problem 5.3.16b
Area functions for constant functions Consider the following functions Ζ and real numbers a (see figure).
(b) Verify that .A'(π) = Ζ(π)
Ζ(t) = 5 , a = -5
Problem 5.3.101b
{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
b) Calculate g'(π)
g(π) = β«βΛ£ sinΒ² t dt
