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문제 5.2.53d
Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.
(a) ∫₀³ 5ƒ(𝓍) d𝓍
문제 5.1.39d
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4
(d) Calculate the midpoint Riemann sum.
문제 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 ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.
문제 5.1.25d
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
f(x) = x + 1 on [0,4]; n = 4
(d) Calculate the left and right Riemann sums.
문제 5.1.27d
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4
(d) Calculate the left and right Riemann sums.
문제 5.1.37d
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
ƒ(𝓍) = 2x + 1 on [0,4] ; n = 4
d) Calculate the midpoint Riemann sum.
문제 5.3.51d
Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)
문제 5.2.55d
Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.
(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍
문제 5.5.15d
Use Table 5.6 to evaluate the following indefinite integrals.
(d) ∫ cos 𝓍/7 d𝓍
문제 5.3.107d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then
B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.
문제 5.3.14d
Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(d) F(8)
문제 5.1.47d
Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(d) 1 + 1/2 + 1/3 + 1/4
문제 5.1.49d
Sigma notation Evaluate the following expressions.
(d) 5
∑ (1 + n²)
n=1
문제 5.2.53.d
Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.
(d) ∫₆³ (ƒ(𝓍) + 2g(𝓍)) d𝓍
문제 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..
∫₀² (𝓍²―2) d𝓍 ; n = 4
문제 5.3.13d
Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(d) F(4)
문제 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.
∫₃⁶ (1―2𝓍) d𝓍 ; n = 6
문제 5.3.87d
Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.
문제 5.2.69d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(d) If ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ is a constant function.
문제 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.
∫₀^π/2 cos 𝓍 d𝓍 ; n = 4
문제 5.1.31d
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6
(d) Calculate the left and right Riemann sums.
문제 5.1.41d
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
ƒ(𝓍) = 1/x on [1,6] ; n = 5
(d) Calculate the midpoint Riemann sum.
문제 5.5.15e
Use Table 5.6 to evaluate the following indefinite integrals.
(e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)
문제 5.1.49e
Sigma notation Evaluate the following expressions.
(e) 3
∑ (2m + 2) / 3
m =1
문제 5.5.15f
Use Table 5.6 to evaluate the following indefinite integrals.
(f) ∫ d𝓍/√36 ―𝓍²
문제 5.1.49f
Sigma notation Evaluate the following expressions.
(f) 3
∑ (3j ― 4)
j =1
문제 5.2.55f
Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.
(f) ∫₄¹ 2f(𝓍) d𝓍
문제 5.3.14g
Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(g) F(2)
