Q: A tank in the shape of a right circular cylinder with radius $$3$$ meters and height $$5$$ meters is filled with water. How much work (in Joules) is required to pump all the water to the top of the tank? (Assume the density of water is $$1000 kg/m$^3$$ and $$g = 9.8 m/s$^2.)$$

$$ W = 1000 \times 9.8 \times \pi \times 3^2$ \times \$$int_0$$^5 (5 - y) \, dy $$

Q: Which of the following is the correct formula for the derivative of the hyperbolic tangent function $$ \tanh x $$?

$$ \dfrac{d}{dx} \tanh x = 1 - \$$tanh^2$$ x $$

Question 1

Which of the following is the correct substitution to evaluate the integral $$ \int 2x \cos(x^2$) \, dx $$?

Accepted Answer

Let $$ u = x^2$ $$, so $$ du = 2x \, dx .$$

Question 2

A tank in the shape of a right circular cylinder with radius $$3$$ meters and height $$5$$ meters is filled with water. How much work (in Joules) is required to pump all the water to the top of the tank? (Assume the density of water is $$1000 kg/m$^3$$ and $$g = 9.8 m/s$^2.)$$

Accepted Answer

$$ W = 1000 	imes 9.8 	imes \pi 	imes 3^2$ 	imes \$$int_0$$^5 (5 - y) \, dy $$

Question 3

Which of the following integrals represents the area between the curves $$ y = x^2$ $$ and $$ y = 2x $$ from $$ x = 0  to$$ $$ x = 2 $$?

Accepted Answer

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

Question 4

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

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

Question 5

A radioactive substance decays according to $$ N(t) = N_0$ e$$^{-kt}$$ . If $$the half-life is $$5730$$ years, what is the value of $$k$$?

Accepted Answer

$$ k = \dfrac{\ln 2}{5730} $$

Question 6

Which of the following is the correct formula for the volume of the solid generated by revolving the region bounded by $$ y = x $$ and $$ y = x^2$ $$ about the $$x-$$axis from $$ x = 0  to$$ $$ x = 1 $$ using the washer method?

Accepted Answer

$$ \$$int_0$$^1 \pi \left[ (x)^2$ - (x^2$)^2$$ \right] \, dx $

Question 7

Evaluate $$ \int x e$$^{x^2}$$ \, dx .$$

Accepted Answer

$$ \dfrac{1}{2} e$$^{x^2}$$ + C $$

Question 8

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

Accepted Answer

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

Question 9

Which of the following is the correct integration by parts formula?

Accepted Answer

$$ \int u \, dv = uv - \int v \, du $$

Question 10

Which of the following is the correct formula for the derivative of the hyperbolic tangent function $$ 	anh x $$?

Accepted Answer

$$ \dfrac{d}{dx} 	anh x = 1 - \$$tanh^2$$ x $$

Question 11

A population of bacteria grows exponentially according to $$ P(t) = P_0$ e$$^{kt}$$ . If $$the population doubles in 3 hours, what is the value of $$k$$?

Accepted Answer

$$ k = \dfrac{\ln 2}{3} $$

Question 12

Which of the following is the correct formula for the Trapezoidal Rule for approximating $$ \int_a^b f(x) \, dx $$ using $$n$$ subintervals?

Accepted Answer

$$ \dfrac{b-a}{2n} \left[ f(x_0$) + 2f(x_1$) + 2f(x_2$) + \cdots + 2f(x_{n-1}) + f(x_n) \right] $$

Question 13

Evaluate $$ \$$int_1$$^e \ln x \, dx .$$

Accepted Answer

$$ e - 1 $$

Question 14

Which of the following is the correct formula for the error bound of the Trapezoidal Rule for $$ \int_a^b f(x) \, dx $$ using $$n$$ subintervals?

Accepted Answer

$$ \dfrac{(b-a)^3$}{12n^2$} \max_{a \leq x \leq b} |f''(x)| $$

Question 15

Which of the following is the correct indefinite integral for $$ \int \dfrac{1}{x^2$ + 1} \, dx $$?

Accepted Answer

$$ \arctan x + C $$

Question 16

Which of the following is the correct formula for the volume of a solid generated by revolving the region bounded by $$ x = 0 $$, $$ x = 1 $$, $$ y = 0 $$, and $$ y = x^2$ $$ about the $$y-$$axis using the shell method?

Accepted Answer

$$ \$$int_0$$^1 2\pi x (x^2$) \, dx $$

Question 17

Which of the following is the correct formula for the derivative of $$ \ln|x| $$?

Accepted Answer

$$ \dfrac{1}{x} $$

Question 18

Which of the following is the correct formula for the integral $$ \int \sinh x \, dx $$?

Accepted Answer

$$ \cosh x + C $$

Question 19

Which of the following is the correct formula for Simpson's Rule for approximating $$ \int_a^b f(x) \, dx $$ using $$n ($$even) subintervals?

Accepted Answer

$$ \dfrac{b-a}{3n} \left[ f(x_0$) + 4f(x_1$) + 2f(x_2$) + \cdots + 4f(x_{n-1}) + f(x_n) \right] $$

Calculus II: Integration and Its Applications – Weeks 1-4 Study Guide

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