Q: Which of the following is the correct integral for the area inside the circle $$ r = 2 in $$polar coordinates?

$$ \$$int_0$$^{2\pi} \$$int_0$$^2 r \, dr \, d\theta $$

Question 1

Which of the following integrals can be most efficiently solved using integration by parts?

Accepted Answer

$$ \int x e^x \, dx $$

Question 2

Evaluate the following integral using trigonometric substitution: $$ \int \frac{dx}{\sqrt{9 - x^2$}} $$

Accepted Answer

$$ \arcsin\left(\frac{x}{3}\right) + C $$

Question 3

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

Accepted Answer

$$ V = \pi \int_a^b [f(x)]^2 dx $$

Question 4

A tank is being filled with water at a rate modeled by $$ R(t) = 5 + 2\sin(t) $$ liters per minute. How much water enters the tank from $$ t = 0  to$$ $$ t = \pi $$ minutes?

Accepted Answer

$$ 5\pi + 4 $$ liters

Question 5

Which of the following series converges?

Accepted Answer

$$ \sum_{n=1}^{\infty} \frac{1}{n^2$} $$

Question 6

Find the Maclaurin series for $$ e^x .$$

Accepted Answer

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

Question 7

Which of the following is the correct parametric equation for a circle of radius $$ r $$ centered at the origin?

Accepted Answer

$$ x = r \cos t, \quad y = r \sin t $$

Question 8

Evaluate $$ \$$int_0$$^1 x e$$^{x^2}$$ dx $$ using substitution.

Accepted Answer

$$ \frac{1}{2}(e^1$ - 1) $$

Question 9

Which of the following is the correct polar equation for a line passing through the pole at an angle $$ \$$theta_0$$ $$?

Accepted Answer

$$ 	heta = \$$theta_0$$ $$

Question 10

A medication is administered at a rate of $$ R(t) = 10e$$^{-0.5t}$$  mg/$$hour. How much medication is administered in the first 2 hours?

Accepted Answer

$$ 20(1 - e$$^{-1}$$)  mg$$

Question 11

Which of the following is the correct formula for the nth Taylor polynomial of $$ f(x) $$ centered at $$ x = a $$?

Accepted Answer

$$ P_n(x) = \sum_{k=0}^n \frac{f$$^{(k)}$$(a)}{k!}(x-a)^k $$

Question 12

Which of the following improper integrals converges?

Accepted Answer

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

Question 13

Given the parametric equations $$ x = t^2$, \ y = t^3$ $$, find $$ \frac{dy}{dx}  in $$terms of $$ t .$$

Accepted Answer

$$ \frac{3t}{2} $$

Question 14

Which of the following is the correct formula for 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 15

A population grows at a rate proportional to its size, $$ \frac{dP}{dt} = 0.2P . If$$ $$ P(0) = 100 $$, what is $$ P(5) $$?

Accepted Answer

$$ 100e$$^{1}$$ $$

Question 16

Which of the following is the correct integral for the area inside the circle $$ r = 2  in $$polar coordinates?

Accepted Answer

$$ \$$int_0$$^{2\pi} \$$int_0$$^2 r \, dr \, d	heta $$

Calculus II Syllabus and Study Guide: Techniques and Applications of Integration, Transcendental Functions, and Series

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