Q: A hospital uses a cylindrical oxygen tank with radius 0.5 meters and height 2 meters. What is the volume of the tank?

$$V = \pi r^2$ h = \pi \cdot (0.5)^2$ \cdot 2 = 0.5\pi\ \text{m}^3$$

Question 1

A pharmaceutical company needs to calculate the work required to pump a liquid medicine from the bottom of a cylindrical tank to the top. The tank is 10 meters tall and has a radius of 2 meters. The medicine has a density of $$ho = 1200\ 	ext{kg/m}^3. $$What is the integral expression for the work required to pump all the medicine to the top of the tank?

Accepted Answer

$$W = \int_{0}^{10} 1200 \cdot \pi \cdot 2^2$ \cdot g \cdot y\ dy$$

Question 2

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

Accepted Answer

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

Question 3

A water tank in the shape of a cone has height 8 meters and base radius 4 meters. If the tank is filled with water (density $$1000\ 	ext{kg/m}^3$$), what is the integral expression for the work required to pump all the water to the top of the tank?

Accepted Answer

$$W = \int_{0}^{8} 1000 \cdot g \cdot \pi \left(\frac{y}{2}\$$right)^2$$ \cdot y\ dy$$

Question 4

Which technique is most appropriate for evaluating the integral $$\int x e$$^{x}$$\ dx$$?

Accepted Answer

Integration by parts

Question 5

A biologist models the population growth of bacteria using the separable differential equation $$\frac{dP}{dt} = kP. $$What is the general solution to this equation?

Accepted Answer

$$P(t) = Ce$$^{kt}$$

Question 6

Which of the following integrals is an improper integral?

Accepted Answer

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

Question 7

A chemical reaction is modeled by the differential equation $$\frac{dy}{dt} = -ky$$, where $$y is $$the concentration of a reactant. If $$y(0) = y_0$, what is the concentration at time t$?

Accepted Answer

$$y(t) = y_0$ e$$^{-kt}$$

Question 8

Which test is most appropriate to determine the convergence of the series $$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$?

Accepted Answer

Comparison test

Question 9

Which of the following is the Maclaurin series for $$e^x$$?

Accepted Answer

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

Question 10

A medical device uses a spring with force $$F(x) = kx. $$What is the work required to stretch the spring from $$x = 0 to$$ $$x = a$$?

Accepted Answer

$$W = \int_{0}^{a} kx\ dx = \frac{1}{2}ka^2$

Question 11

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

Accepted Answer

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

Question 12

Evaluate the improper integral $$\int_{1}^{\infty} \frac{1}{x^p}\ dx$$ for $$p \u003e 1. $$What is the result?

Accepted Answer

$$\frac{1}{p-1}$$

Question 13

Which of the following is the correct partial fraction decomposition for $$\frac{3x+2}{x^2$+x}$$?

Accepted Answer

$$\frac{3x+2}{x^2$+x} = \frac{3}{x} + \frac{-1}{x+1}$$

Question 14

A power series $$\sum_{n=0}^{\infty} a_n x^n$$ converges for $$|x| \u003c R. $$What is $$R$$ called?

Accepted Answer

Radius of convergence

Question 15

Which of the following is the Taylor series for $$\sin x$$ centered at $$x = 0$$?

Accepted Answer

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

Question 16

A hospital uses a cylindrical oxygen tank with radius 0.5 meters and height 2 meters. What is the volume of the tank?

Accepted Answer

$$V = \pi r^2$ h = \pi \cdot (0.5)^2$ \cdot 2 = 0.5\pi\ 	ext{m}^3$$

Question 17

Which of the following is the correct formula for the center of mass $$\bar{x} of a $$thin rod of length $$L$$ with density function $$\rho(x)$$?

Accepted Answer

$$\bar{x} = \frac{\int_{0}^{L} x \rho(x)\ dx}{\int_{0}^{L} \rho(x)\ dx}$$

Question 18

A patient receives a medication whose concentration in the bloodstream decreases exponentially according to $$C(t) = C_0$ e$$^{-kt}$$. If C_0$ = 50 mg/L $$and $$k = 0.2 hr$^{-1}$$, what is the concentration after 5 hours?

Accepted Answer

$$C(5) = 50 e$$^{-1}$$

Question 19

Which of the following is the correct formula for the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$A = \int_{a}^{b} [f(x) - g(x)]\ dx$$

Calculus II (MATH 242) Syllabus and Course Structure: Chapters 6–9 Study Guide

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