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Problem 5.5.15d
Use Table 5.6 to evaluate the following indefinite integrals.
(d) β« cos π/7 dπ
Problem 5.1.71d
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(d) Assuming the velocity remains 10 m/s, for t β₯ 5, find the function that gives the displacement between t = 0 and any time t β₯ 5.
Problem 5.1.31d
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} Ζ(π) = e Λ£/β on [1,4]; n = 6
(d) Calculate the left and right Riemann sums.
Problem 5.3.107d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(d) If A(π) = 3πΒ²β πβ 3 is an area function for Ζ, then
B(π) = 3πΒ² β π is also an area function for Ζ.
Problem 5.1.27d
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.β
{Use of Tech} Ζ(π) = cos π on [0. Ο/2]; n = 4
(d) Calculate the left and right Riemann sums.
Problem 5.1.49d
Sigma notation Evaluate the following expressions.
(d) 5
β (1 + nΒ²)
n=1
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 5.3.14d
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«βΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(d) F(8)
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 5.2.69d
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 5.1.29d
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
Ζ(π) = xΒ² β 1 on [2,4]; n = 4
(d) Calculate the left and right Riemann sums.
Problem 5.5.16d
Use Table 5.6 to evaluate the following definite integrals.
(d) β«β^Ο/ΒΉβΆ sec Β² 4π dπ
Problem 5.1.39d
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} Ζ(π) = βx on [1,3] ; n = 4
(d) Calculate the midpoint Riemann sum.
Problem 5.1.41d
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.β
Ζ(π) = 1/x on [1,6] ; n = 5
(d) Calculate the midpoint Riemann sum.
Problem 5.3.87d
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 5.2.32d
{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 5.1.47d
Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(d) 1 + 1/2 + 1/3 + 1/4
Problem 5.1.37d
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.β
Ζ(π) = 2x + 1 on [0,4] ; n = 4
d) Calculate the midpoint Riemann sum.
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 5.3.13d
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«ββΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(d) F(4)
Problem 5.1.25d
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.β
f(x) = x + 1 on [0,4]; n = 4
(d) Calculate the left and right Riemann sums.
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 5.5.15e
Use Table 5.6 to evaluate the following indefinite integrals.
(e) β« dπ/(81 + 9πΒ²) (Hint: Factor a 9 out of the denominator first.)
Problem 5.1.49e
Sigma notation Evaluate the following expressions.
(e) 3
β (2m + 2) / 3
m =1
Problem 5.5.15f
Use Table 5.6 to evaluate the following indefinite integrals.
(f) β« dπ/β36 βπΒ²
Problem 5.1.49f
Sigma notation Evaluate the following expressions.
(f) 3
β (3j β 4)
j =1
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 5.3.14g
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«βΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(g) F(2)
