Question 1

Which of the following is the correct partial fraction decomposition for the rational function $$\frac{2x+3}{(x-1)(x+2)}$$?

Accepted Answer

$$\frac{2x+3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$

Question 2

Given $$\int_{0}^{1} \pi x^2$ dx$$, what real-world scenario does this integral represent?

Accepted Answer

The volume of a solid formed by rotating the region under $$y = x$$ from $$x = 0 to$$ $$x = 1$$ about the x-axis.

Question 3

What is the correct formula for the volume of a solid of revolution generated by rotating a region about the x-axis using the disk method?

Accepted Answer

$$V = \int_{a}^{b} \pi [R(x)]^2 dx$$

Question 4

If $$Q(x)$$ contains a repeated linear factor $$(x - r)^n$$, what is the correct form for the partial fraction decomposition?

Accepted Answer

$$\frac{A_1$}{x - r} + \frac{A_2$}{(x - r)^2$} + \cdots + \frac{A_n}{(x - r)^n}$$

Question 5

A solid is formed by rotating the region bounded by $$y = x^3$, y = 8$, and x = 0$ about the y-axis. What is the radius of a typical disk at height y$?

Accepted Answer

$$x = y$$^{1/3}$$

Question 6

Which of the following is the correct formula for the area of a washer cross-section when rotating the region between $$y = x$$ and $$y = x^2$ about the x-axis?

Accepted Answer

$$A(x) = \pi [x^2$ - x^4$]$$

Question 7

When integrating $$\frac{1}{x^2$ + 4}$$, which substitution is most useful for expressing the result in terms of $$	an$$^{-1}$$?

Accepted Answer

Let $$u = x/2 so $$that $$\int \frac{1}{x^2$ + 4} dx = \frac{1}{2} 	an$$^{-1}$$(x/2) + C$$

Question 8

Which principle states that two solids with equal cross-sectional areas at every level have equal volumes?

Accepted Answer

Cavalieri's principle

Question 9

For the rational function $$\frac{x}{(x-2)(x^2$+1)(x^2+4$$)}$, what is the correct form of its partial fraction decomposition?

Accepted Answer

$$\frac{A}{x-2} + \frac{Bx+C}{x^2$+1} + \frac{Dx+E}{x^2$+4}$$

Question 10

What is the volume of a sphere of radius $$r$$ using the disk method?

Accepted Answer

$$V = \int_{-r}^{r} \pi (r^2$ - x^2$) dx = \frac{4}{3} \pi r^3$

Question 11

When integrating $$\frac{1}{x^2$ - a^2$}$$, which partial fraction decomposition should be used?

Accepted Answer

$$\frac{1}{x^2$ - a^2$} = \frac{1}{2a} \left( \frac{1}{x-a} - \frac{1}{x+a} \right)$$

Question 12

A region bounded by $$y = x$$ and $$y = x^2$ is rotated about the line y = 2$. What is the formula for the area of a typical washer cross-section?

Accepted Answer

$$A(x) = \pi [(2-x^2$)^2$$ - $$(2-x)^2$$]$

Question 13

Which step is NOT part of the standard process for calculating the volume of a solid using cross-sections?

Accepted Answer

Multiply the area by the circumference

Question 14

If $$Q(x)$$ contains an irreducible quadratic factor $$(x^2 + px + q)^n$, what is the correct form for the partial fraction decomposition?

Accepted Answer

$$\frac{Bx + C}{x^2$ + px + q} + \frac{B_2x + C_2$}{(x^2$ + px + q)^2$} + \cdots + \frac{B_nx + C_n}{(x^2$ + px + q)^n}$$

Question 15

Which of the following is a correct application of the washer method for finding the volume of a solid of revolution?

Accepted Answer

$$V = \int_{a}^{b} \pi ([R(x)]^2 - [r(x)]^2) dx$$

Question 16

If a rational function is improper (degree of numerator $$\geq$$ degree of denominator), what is the first step in integrating it?

Accepted Answer

Perform long division to obtain a proper fraction

Question 17

What is the volume of the solid obtained by rotating the region under $$y = \sqrt{x}$$ from $$x = 0 to$$ $$x = 1$$ about the x-axis?

Accepted Answer

$$V = \int_{0}^{1} \pi x dx = \frac{\pi}{2}$$

Integration of Rational Functions by Partial Fractions and Volumes Using Cross-Sections

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