Q: Which of the following is a necessary condition for the convergence of the improper integral $$\int_a^b f(x)\,dx$$ where $$f(x) is $$discontinuous at $$c$$ with $$a \u003c c \u003c b$$?

The limit $$\lim_{t \to c^-} \int_a^t f(x)\,dx$$ and $$\lim_{t \to c^+} \int_t^b f(x)\,dx$$ must both exist and be finite.

Q: Suppose $$f(x) is $$continuous on $$[a, b)$$ but has an infinite discontinuity at $$b. $$How is $$\int_a^b f(x)\,dx$$ defined?

$$\int_a^b f(x)\,dx = \lim_{t \to b^-} \int_a^t f(x)\,dx$$

Q: Suppose $$f(x) is $$continuous on $$[0, 2]$$ except at $$x = 1$$, where it has an infinite discontinuity. Which of the following is the correct way to write $$\$$int_0$$^2 f(x)\,dx$$?

$$\$$int_0$$^2 f(x)\,dx = \lim_{t \to 1^-} \$$int_0$$^t f(x)\,dx + \lim_{t \to 1^+} \int_t^2$ f(x)\,dx$$

Question 1

Which of the following statements best describes the Fundamental Theorem of Calculus (FTC)?

Accepted Answer

It connects differentiation and integration by stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x)\,dx = F(b) - F(a).$$

Question 2

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$F(x) is an $$antiderivative of $$f(x). $$What is the value of $$\int_a^b f(x)\,dx$$?

Accepted Answer

$$F(b) - F(a)$$

Question 3

Evaluate the improper integral $$\$$int_1$$^\infty \frac{1}{x^2$}\,dx.$$

Accepted Answer

$$1$$

Question 4

Which of the following is a necessary condition for the convergence of the improper integral $$\int_a^b f(x)\,dx$$ where $$f(x) is $$discontinuous at $$c$$ with $$a \u003c c \u003c b$$?

Accepted Answer

The limit $$\lim_{t 	o c^-} \int_a^t f(x)\,dx$$ and $$\lim_{t 	o c^+} \int_t^b f(x)\,dx$$ must both exist and be finite.

Question 5

Evaluate $$\$$int_0$$^1 \frac{1}{\sqrt{x}}\,dx.$$

Accepted Answer

$$2$$

Question 6

Which of the following integrals diverges?

Accepted Answer

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

Question 7

A tank is being filled with water at a rate given by $$r(t) = 3t^2$ liters per minute. How much water enters the tank between t = 0$ and t = 2$ minutes?

Accepted Answer

$$\$$int_0$$^2 3t^2$\,dt = 8$$ liters

Question 8

Which of the following statements is TRUE about the convergence of $$\$$int_0$$^1 \frac{1}{x^p}\,dx$$?

Accepted Answer

Question 9

Suppose $$f(x) is $$continuous on $$[a, b)$$ but has an infinite discontinuity at $$b. $$How is $$\int_a^b f(x)\,dx$$ defined?

Accepted Answer

$$\int_a^b f(x)\,dx = \lim_{t 	o b^-} \int_a^t f(x)\,dx$$

Question 10

Evaluate $$\$$int_0$$^\infty e$$^{-x}$$\,dx.$$

Accepted Answer

$$1$$

Question 11

Which of the following is an example of an improper integral due to an infinite interval?

Accepted Answer

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

Question 12

A medication is absorbed into the bloodstream at a rate $$r(t) = 5e$$^{-0.5t}$$ mg/hour. How much medication is absorbed in the first 4 hours?

Accepted Answer

$$\$$int_0$$^4 5e$$^{-0.5t}$$\,dt = 10(1 - e$$^{-2}$$) mg$$

Question 13

Which of the following best describes when an improper integral $$\int_a^b f(x)\,dx is $$said to diverge?

Accepted Answer

When the corresponding limit does not exist or is infinite.

Question 14

Evaluate $$\$$int_0$$^1 \ln(x)\,dx.$$

Accepted Answer

$$-1$$

Question 15

Which of the following is a correct statement about the convergence of $$\$$int_1$$^\infty \frac{1}{x^p}\,dx$$?

Accepted Answer

It converges if and only if $$p \u003e 1.$$

Question 16

Suppose $$f(x) is $$continuous on $$[0, 2]$$ except at $$x = 1$$, where it has an infinite discontinuity. Which of the following is the correct way to write $$\$$int_0$$^2 f(x)\,dx$$?

Accepted Answer

$$\$$int_0$$^2 f(x)\,dx = \lim_{t 	o 1^-} \$$int_0$$^t f(x)\,dx + \lim_{t 	o 1^+} \int_t^2$ f(x)\,dx$$

Question 17

A radioactive substance decays at a rate $$r(t) = ke$$^{-kt}$$ grams per hour. What is the total amount decayed from t = 0$ to t$$ = \infty$$?

Accepted Answer

Improper Integrals and Convergence in Calculus

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