Q: A tank in the shape of a 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 $$1000 \ kg/m^3$ and g = 9.8$$ \ $$m/s^2.)$$

$$ W = 1000 \times 9.8 \times \pi \times 3^2$ \times 5^2$ $$

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 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

$$ W = 1000 	imes 9.8 	imes \pi 	imes 3^2$ 	imes 5^2$ $$

Question 3

Evaluate the definite integral $$ \$$int_0$$^1 e$$^{2x}$$ \, dx .$$

Accepted Answer

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

Question 4

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

Accepted Answer

$$ \$$int_0$$^2 [(x + 2) - x^2$] \, dx $$

Question 5

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 6

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

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

Question 7

Which of the following integrals can be evaluated using integration by parts?

Accepted Answer

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

Question 8

Which of the following is the correct formula for the volume of the solid generated by revolving the region bounded by $$ y = f(x) $$ and the $$x-$$axis from $$ x = a  to$$ $$ x = b $$ about the $$x-$$axis using the disk method?

Accepted Answer

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

Question 9

Which of the following is the correct derivative of the hyperbolic sine function $$ \sinh x $$?

Accepted Answer

$$ \cosh x $$

Question 10

A force $$ F(x) = 5x $$ Newtons moves an object along a straight path from $$ x = 0  to$$ $$ x = 4 $$ meters. What is the total work done?

Accepted Answer

$$ 40 	ext{ Joules} $$

Question 11

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

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

Question 12

Which of the following is the correct integral representation 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 13

Which of the following is the correct result for $$ \int \ln x \, dx $$?

Accepted Answer

$$ x \ln x - x + C $$

Question 14

Which of the following is the correct formula for the volume of a solid of revolution using the shell method about the $$y-$$axis?

Accepted Answer

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

Question 15

Which of the following is the correct integral for the area between $$ y = e^x $$ and $$ y = x $$ from $$ x = 0  to$$ $$ x = 1 $$?

Accepted Answer

$$ \$$int_0$$^1 (e^x - x) \, dx $$

Question 16

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

Accepted Answer

$$ \frac{1}{x} $$

Question 17

Which of the following is the correct formula for the error bound of the Trapezoidal Rule for $$ n $$ subintervals on $$ [a, b] $$?

Accepted Answer

$$ \frac{(b-a)^3$}{12n^2$} \max |f''(x)| $$

Question 18

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

Accepted Answer

$$ \arctan x + C $$

Calculus II: Integration and Its Applications – Study Notes (Weeks 1–4)

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