Question

Suppose that f and g are two continuous functions with f(x) ? g(x) ? 0 for a ? x ? b, and ??? f(x) dx, ??? g(x) dx are two second type improper integrals. Which one of the following statements is correct? Choose all correct ones. a. If ??? f(x) dx is divergent, then ??? g(x) dx is convergent. b. If ??? f(x) dx is convergent, then ??? g(x) dx is convergent. c. If ??? f(x) dx is divergent, then the convergence of ??? g(x) dx is indetermined. d. If ??? f(x) dx is divergent, then ??? g(x) dx is divergent. e. If ??? f(x) dx is convergent, then the convergence of ??? g(x) dx is indetermined.

          Suppose that f and g are two continuous functions with f(x) ? g(x) ? 0 for a ? x ? b, and ??? f(x) dx, ??? g(x) dx are two second type improper integrals. Which one of the following statements is correct? Choose all correct ones.

a. If ??? f(x) dx is divergent, then ??? g(x) dx is convergent.

b. If ??? f(x) dx is convergent, then ??? g(x) dx is convergent.

c. If ??? f(x) dx is divergent, then the convergence of ??? g(x) dx is indetermined.

d. If ??? f(x) dx is divergent, then ??? g(x) dx is divergent.

e. If ??? f(x) dx is convergent, then the convergence of ??? g(x) dx is indetermined.

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Suppose that f and g are two continuous functions with f(x) ? g(x) ? 0 for a ? x ? b, and ??? f(x) dx, ??? g(x) dx are two second type improper integrals. Which one of the following statements is correct? Choose all correct ones.

a. If ??? f(x) dx is divergent, then ??? g(x) dx is convergent.

b. If ??? f(x) dx is convergent, then ??? g(x) dx is convergent.

c. If ??? f(x) dx is divergent, then the convergence of ??? g(x) dx is indetermined.

d. If ??? f(x) dx is divergent, then ??? g(x) dx is divergent.

e. If ??? f(x) dx is convergent, then the convergence of ??? g(x) dx is indetermined.

Added by Angela F.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Andrew Noble

03/05/2023

Step 1

If $\int_a^b f(x) dx$ is divergent, then $\int_a^b g(x) dx$ is also divergent. This is because $f(x) \geq g(x) \geq 0$ for all $x \in (a, b)$. If the integral of the larger function $f(x)$ is divergent, then the integral of the smaller function $g(x)$ must also be Show more…

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Suppose that f and g are two continuous functions with f(x) ≥ g(x) ≥ 0 for a ≤ x ≤ b, and ∫ₐ၇ f(x) dx, ∫ₐ၇ g(x) dx are two second type improper integrals. Which one of the following statements is correct? Choose all correct ones. a. If ∫ₐ၇ f(x) dx is divergent, then ∫ₐ၇ g(x) dx is convergent. b. If ∫ₐ၇ f(x) dx is convergent, then ∫ₐ၇ g(x) dx is convergent. c. If ∫ₐ၇ f(x) dx is divergent, then the convergence of ∫ₐ၇ g(x) dx is indetermined. d. If ∫ₐ၇ f(x) dx is divergent, then ∫ₐ၇ g(x) dx is divergent. e. If ∫ₐ၇ f(x) dx is convergent, then the convergence of ∫ₐ၇ g(x) dx is indetermined.

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