Question 1

Evaluate the following definite integral using an appropriate substitution: 
$$
\$$int_0$$^1 x \sqrt{1 - x^2$} \, dx
$$

Accepted Answer

$$\frac{1}{3}$$

Question 2

A tank in the shape of a solid of revolution is formed by revolving $$y = x^2$ for 0$$ \leq x \leq 2$$ about the y$-axis. If the tank is filled with salt water (density 73$ lb/ft^3$), which integral represents the work required to pump all the water to the top of the tank?

Accepted Answer

$$\$$int_0$$^2 73 \cdot 2\pi x \cdot x^2$ \cdot (4 - x^2$) \, dx$$

Question 3

Which of the following series converges by the Alternating Series Test?

Accepted Answer

$$\sum_{n=1}^\infty \frac{(-1)^n \ln n}{n}$$

Question 4

Find the area of the region bounded by $$y = x^2$ and y = 3x$.

Accepted Answer

$$\$$int_0$$^3 (3x - x^2$) \, dx$$

Question 5

Evaluate the following integral using trigonometric substitution: 
$$
\int \frac{1}{\sqrt{81 - x^2$}} \, dx
$$

Accepted Answer

$$\arcsin\left(\frac{x}{9}\right) + C$$

Question 6

Which of the following is the correct setup for the volume of the solid formed by rotating the region between $$y = x^2$ and y = 3x$ about the y$-axis using the method of cylindrical shells?

Accepted Answer

$$2\pi \$$int_0$$^3 x (3x - x^2$) \, dx$$

Question 7

Find the Maclaurin series for $$f(x) = e$$^{3x}$$.

Accepted Answer

$$\sum_{n=0}^\infty \frac{(3x)^n}{n!}$$

Question 8

Which of the following is the correct reduction formula for $$\int \cos^n x \, dx$$?

Accepted Answer

$$\int \cos^n x \, dx = \frac{\cos$$^{n-1}$$ x \sin x}{n} + \frac{n-1}{n} \int \cos$$^{n-2}$$ x \, dx$$

Question 9

Determine the radius of convergence for the power series $$\sum_{n=1}^\infty \frac{(n!)^2$ x^n}{(3n-1)!}.$$

Accepted Answer

$$3$$

Question 10

Evaluate the following integral using integration by parts: 
$$
\int x^2$ e^x \, dx
$$

Accepted Answer

$$e^x(x^2$ - 2x + 2) + C$$

Question 11

Which of the following is the correct Taylor series for $$f(x) = x^2$ \ln x$$ centered at $$a = 1$$?

Accepted Answer

$$1^2$$ \ln 1 + 2 \cdot 1 \ln 1 (x-1) + (1 + 2 \ln $$1)(x-1)^2$$ + \cdots$$

Question 12

Which of the following integrals can be evaluated using the method of partial fractions?

Accepted Answer

$$\int \frac{x+2}{x(x^2$+4)} \, dx$$

Question 13

Find the arc length of the curve $$y = \frac{2}{3} x$$^{3/2}$$ for 0$$ \leq x \leq 1$$.

Accepted Answer

$$\frac{8}{27} \left[ (1 + 1$$)^{3/2}$$ - 0 \right]$$

Question 14

Which of the following is the correct Maclaurin series for $$\sin x$$?

Accepted Answer

$$\sum_{n=0}^\infty \frac{(-1)^n x$$^{2n+1}$$}{(2n+1)!}$$

Question 15

Which of the following is the correct setup for the area of the region bounded by $$x = y^2$ and y = x - 2$?

Accepted Answer

$$\int_{y_1$}^{y_2$} [(y + 2) - y^2$] \, dy$$

Question 16

Which of the following series is a geometric series?

Accepted Answer

$$1 + \frac{1}{\pi} + \frac{1}{\pi^2$} + \frac{1}{\pi^3$} + \cdots$$

Question 17

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

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

Question 18

Which acronym is commonly used to help choose $$u in $$integration by parts, and what does it stand for?

Accepted Answer

ILATE: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential

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