Q: Which of the following is the correct definition for a Type I improper integral?

$$\int_a^\infty f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx$$

Q: Which of the following is the correct definition for a Type II improper integral?

$$\int_a^b f(x) dx = \lim_{c \to a^+} \int_c^b f(x) dx$$

Question 1

Which of the following integrals is convergent?

Accepted Answer

$$\$$int_1$$^\infty \frac{1}{x^2$} dx$$

Question 2

Evaluate $$\$$int_1$$^t \frac{1}{x^2$} dx$$ and find the area under $$y = \frac{1}{x^2$}$$ from $$x = 1 to$$ $$x = t.$$

Accepted Answer

$$1 - \frac{1}{t}$$

Question 3

For what values of $$p$$ does the improper integral $$\$$int_1$$^\infty \frac{1}{x^p} dx$$ converge?

Accepted Answer

$$p \u003e 1$$

Question 4

Which statement best describes the area under $$y = \frac{1}{x^2$}$$ from $$x = 1 to$$ $$x = \infty$$?

Accepted Answer

The area is equal to 1.

Question 5

Which of the following is the correct definition for a Type I improper integral?

Accepted Answer

$$\int_a^\infty f(x) dx = \lim_{b 	o \infty} \int_a^b f(x) dx$$

Question 6

Evaluate $$\int_{-\infty}^{\infty} \frac{1}{1 + x^2$} dx.$$

Accepted Answer

$$\pi$$

Question 7

Which of the following integrals is divergent due to a vertical asymptote within the interval?

Accepted Answer

$$\$$int_0$$^3 \frac{1}{x-1} dx$$

Question 8

Suppose $$f(x)$$ and $$g(x)$$ are continuous on $$[a, \infty)$$ and $$0 \leq f(x) \leq g(x)$$ for all $$x \geq a. If$$ $$\int_a^\infty g(x) dx$$ diverges, what can be concluded?

Accepted Answer

$$\int_a^\infty f(x) dx$$ also diverges.

Question 9

Which theorem allows you to determine the convergence of an improper integral by comparing it to another function?

Accepted Answer

Direct Comparison Test

Question 10

Evaluate $$\$$int_0$$^1 \ln x dx.$$

Accepted Answer

$$-1$$

Question 11

Which of the following statements about improper integrals is correct?

Accepted Answer

If the limit exists and is finite, the improper integral converges.

Question 12

Which integral is convergent by the comparison theorem if $$f(x) = e$$^{-x^2}$$ and g(x$$) = $$e^{-x}$$ for $$x \geq 1$$?

Accepted Answer

$$\$$int_1$$^\infty e$$^{-x^2}$$ dx$$

Question 13

Which of the following is the correct definition for a Type II improper integral?

Accepted Answer

$$\int_a^b f(x) dx = \lim_{c 	o a^+} \int_c^b f(x) dx$$

Question 14

If $$\$$int_1$$^\infty \frac{1}{x^p} dx$$ diverges, what can be said about $$\$$int_1$$^\infty \frac{1}{x} dx$$?

Accepted Answer

It also diverges.

Question 15

Which of the following is the correct value for $$\$$int_1$$^2 \frac{1}{x^2$} dx$$?

Accepted Answer

$$\frac{1}{2}$$

Question 16

What is the area under $$y = \frac{1}{x^2$}$$ from $$x = 1 to$$ $$x = 3$$?

Accepted Answer

$$\frac{2}{3}$$

Question 17

Which integral is divergent due to the behavior of the integrand as $$x 	o \infty$$?

Accepted Answer

$$\$$int_1$$^\infty \frac{1}{x} dx$$

Question 18

If $$f(x) is $$continuous on $$[a, \infty)$$ and $$\int_a^\infty f(x) dx$$ converges, what does this imply about the area under $$f(x)$$ from $$x = a to$$ $$x = \infty$$?

Accepted Answer

The area is finite.

Question 19

Which of the following is the correct value for $$\$$int_1$$^5 \frac{1}{x^2$} dx$$?

Accepted Answer

$$\frac{4}{5}$$

Improper Integrals: Infinite Intervals and Discontinuous Integrands

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