Q: Which of the following statements about the convergence of an improper integral $$\int_a^\infty f(x) \, dx is $$TRUE?

The integral converges if $$\lim_{b \to \infty} \int_a^b f(x) \, dx$$ exists and is finite.

Question 1

Which of the following is the correct formula for the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$, where $$f(x) \geq g(x) on$$ $$[a, b]$$?

Accepted Answer

$$A = \int_a^b [f(x) - g(x)] \, dx$$

Question 2

Suppose you are asked to find the area enclosed between $$y = 2x - x^2$ and y = x$. What is the first step you should take?

Accepted Answer

Set $$2x - x^2$ = x$$ and solve for $$x to $$find the points of intersection.

Question 3

Evaluate $$\displaystyle \$$int_0$$^\pi \sin^2$ x \, dx.$$

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 4

Which of the following statements about the convergence of an improper integral $$\int_a^\infty f(x) \, dx is $$TRUE?

Accepted Answer

The integral converges if $$\lim_{b 	o \infty} \int_a^b f(x) \, dx$$ exists and is finite.

Question 5

Find the area of the region bounded by $$y = \sin x$$ and $$y = \cos x$$ between $$x = 0$$ and $$x = \frac{\pi}{2}.$$

Accepted Answer

$$2 - \sqrt{2}$$

Question 6

Which of the following is the correct formula for the arc length of a curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$$

Question 7

A region is bounded above by $$y = x^2$ and below by y = 0$ from x = 0$ to x = 2$. What is the area of this region?

Accepted Answer

$$\dfrac{8}{3}$$

Question 8

Which of the following integrals represents the area between $$y = e^x$$ and $$y = x$$ from $$x = 0 to$$ $$x = 1$$?

Accepted Answer

$$\$$int_0$$^1 [e^x - x] \, dx$$

Question 9

Suppose $$\$$int_1$$^\infty \dfrac{1}{x^p} \, dx$$ converges. What is the range of $$p$$ for which this is true?

Accepted Answer

$$p \u003e 1$$

Question 10

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

Accepted Answer

$$8$$ liters

Question 11

Which of the following is the correct setup for the area between $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$?

Accepted Answer

$$\$$int_0$$^2 [2x - x^2$] \, dx$$

Question 12

Evaluate $$\displaystyle \$$int_0$$^1 x e$$^{x^2}$$ \, dx.$$

Accepted Answer

$$\dfrac{1}{2}(e - 1)$$

Question 13

Which of the following is the correct formula for the length of the curve $$x = g(y)$$ from $$y = c to$$ $$y = d$$?

Accepted Answer

$$L = \int_c^d \sqrt{1 + [g'(y)]^2} \, dy$$

Question 14

A region is bounded by $$y = x^2$ and y = 4$. What are the x$-values where these curves intersect?

Accepted Answer

$$x = -2$$ and $$x = 2$$

Question 15

Which of the following integrals represents the area between $$y = \sin x$$ and $$y = \cos x$$ from $$x = 0 to$$ $$x = \frac{\pi}{2}$$?

Accepted Answer

$$\$$int_0$$^{\frac{\pi}{2}} |\sin x - \cos x| \, dx$$

Question 16

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(x) \geq 0$$ for all $$x in$$ $$[a, b]$$, what does $$\int_a^b f(x) \, dx$$ represent geometrically?

Accepted Answer

The area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$ above the $$x-$$axis.

Question 17

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

Accepted Answer

$$-1$$

Question 18

Which of the following is the correct formula for the area between two curves $$x = f(y)$$ and $$x = g(y)$$ from $$y = c to$$ $$y = d$$, where $$f(y) \geq g(y)$$?

Accepted Answer

$$A = \int_c^d [f(y) - g(y)] \, dy$$

Applications of Integration: Area Between Curves and Related Concepts

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