Question 1

Which of the following integrals gives the area enclosed by the curve $$y = x^2$ and the line y = x + 2$?

Accepted Answer

$$\int_{a}^{b} [(x + 2) - x^2$] dx$$

Question 2

Suppose you want to find the volume of the solid generated by revolving the region bounded by $$y = x^2$, y = 0$, and x = 2$ about the y$-axis. Which method is most appropriate?

Accepted Answer

Shell method integrating with respect to $$x$$

Question 3

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

Accepted Answer

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

Question 4

Evaluate $$\int \frac{2x + 1}{\sqrt{x^2$ + x}} dx$$ using substitution. What is the most appropriate substitution?

Accepted Answer

$$u = x^2$ + x$$

Question 5

The region bounded by $$y = \sqrt{x}$$, $$y = x/2$$, and the $$y-$$axis is revolved about the $$x-$$axis. Which integral represents the volume of the solid generated?

Accepted Answer

$$2\pi \int_{0}^{2} x (\sqrt{x} - x/2) dx$$

Question 6

Which of the following is the correct antiderivative for $$\int \sec^2$(x) dx$$?

Accepted Answer

$$	an(x) + C$$

Question 7

Find the area of the region bounded by $$y = x^2$ and y = 2x$ for x$ between 0$ and 2$.

Accepted Answer

$$\int_{0}^{2} [2x - x^2$] dx$$

Question 8

Which of the following integrals gives the surface area of the solid generated by revolving $$y = f(x)$$ about the $$x-$$axis from $$x = a to$$ $$x = b$$?

Accepted Answer

$$S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + [f'(x)]^2} dx$$

Question 9

Let $$F(x) be an $$antiderivative of $$f(x) = \frac{1}{x}. $$Which of the following expresses $$\int_{a}^{b} \frac{1}{x} dx in $$terms of $$F$$?

Accepted Answer

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

Question 10

Which substitution is most appropriate for evaluating $$\int \frac{1}{\sqrt{1 - x^2$}} dx$$?

Accepted Answer

$$x = \sin 	heta$$

Question 11

A region is bounded by $$y = x^2$, y = 0$, and x = 1$. What is the volume of the solid generated by revolving this region about the x$-axis?

Accepted Answer

$$\pi \int_{0}^{1} (x^2$)^2 dx$

Question 12

Which of the following is the correct formula for the length of the curve $$y = x^3$ on the interval 1$$ \leq x \leq 2$$?

Accepted Answer

$$\int_{1}^{2} \sqrt{1 + 9x^4$} dx$$

Question 13

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

Accepted Answer

$$\int_{0}^{\frac{\pi}{2}} [\cos x - \sin x] dx$$

Question 14

The region bounded by $$y = x^2$, y = 4$, and x = 0$ is revolved about the y$-axis. Which integral gives the volume of the solid generated?

Accepted Answer

$$2\pi \int_{0}^{2} x (4 - x^2$) dx$$

Question 15

Which of the following is the correct antiderivative for $$\int 	an x dx$$?

Accepted Answer

$$-\ln|\cos x| + C$$

Question 16

Let $$R be $$the region bounded by $$y = x^2$ and y = x$. Which of the following integrals gives the area of R$ between x = 0$ and x = 1$?

Accepted Answer

$$\int_{0}^{1} [x - x^2$] dx$$

Question 17

Which of the following is the correct formula for the surface area of the solid generated by revolving $$y = f(x)$$ about the $$y-$$axis from $$x = a to$$ $$x = b$$?

Accepted Answer

$$S = 2\pi \int_{a}^{b} x \sqrt{1 + [f'(x)]^2} dx$$

Question 18

Which of the following is the correct antiderivative for $$\int \frac{1}{x^2$ + 1} dx$$?

Accepted Answer

$$\arctan x + C$$

Calculus II Study Guide: Integration Techniques, Area, Volume, and Applications

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