Question 1

A 5-lb bucket is lifted from the ground to the top of a 20-ft building using a rope that weighs 0.08 lb/ft, as shown in the image. How much work is done in lifting both the bucket and the rope to the top of the building?

Accepted Answer

$$ W = \$$int_0$$^{20} 5 \, dx + \$$int_0$$^{20} 0.08(20-x) \, dx = 100 + 16 = 116 	ext{ ft-lb} $$

Question 2

Evaluate the integral $$ \int \sin^2$ x \, dx . $$Which of the following is the correct antiderivative?

Accepted Answer

$$ \frac{x}{2} - \frac{1}{4} \sin 2x + C $$

Question 3

Find the volume of the solid whose cross-sections perpendicular to the $$x-$$axis from $$x = -1 to$$ $$x = 1$$ are disks with diameters running from $$y = x^2$ to y = 2$$ - $$x^2.$$

Accepted Answer

$$ V = \int_{-1}^{1} \pi \left( (2 - x^2$) - x^2$ \$$right)^2$$ dx $$

Question 4

Evaluate $$ \int_{0}^{\pi} \frac{\sin x}{2 - \cos x} dx .$$

Accepted Answer

$$ 2 \ln 2 $$

Question 5

Find the total area of the region bounded by $$ x = 12y^2$ - 12y^3$ $$ and $$ x = 2y^2$ - 2y $$ for $$ 0 \leq y \leq 1 .$$

Accepted Answer

$$ \$$int_0$$^1 |(12y^2$ - 12y^3$) - (2y^2$ - 2y)| dy = \$$int_0$$^1 (10y^2$ - 12y^3$ + 2y) dy $$

Question 6

A bag of sand originally weighing 144 lb is lifted 18 ft, losing sand at a constant rate so that half the sand is gone at the top. What is the total work done (neglecting the bag's weight)?

Accepted Answer

$$ W = \$$int_0$$^{18} (144 - 4x) dx $$

Question 7

Find the length of the curve $$ x = \frac{y^3$}{6} + \frac{1}{2y} $$ from $$ y = 2  to$$ $$ y = 3 .$$

Accepted Answer

$$ L = \$$int_2$$^3 \sqrt{1 + \left( \frac{d}{dy} \left( \frac{y^3$}{6} + \frac{1}{2y} \right) \$$right)^2$$ } dy $$

Question 8

A semicircular plate of radius 5 ft rests vertically on its diameter at the bottom of a pool filled with water to a depth of 6 ft. What is the fluid force on one side of the plate? (Water weighs 62.4 lb/ft$$^3.)$$

Accepted Answer

$$ F = 62.4 \$$int_0$$^5 (6 - y) 2\sqrt{25 - y^2$} dy $$

Question 9

Find the surface area generated by revolving the segment $$ y = \sqrt{x} $$ for $$ \frac{3}{4} \leq x \leq \frac{15}{4} $$ about the $$x-$$axis.

Accepted Answer

$$ S = 2\pi \int_{3/4}^{15/4} \sqrt{x} \sqrt{1 + \left( \frac{1}{2\sqrt{x}} \$$right)^2$$ } dx $$

Question 10

Find the center of mass $$(\bar{x}, \bar{y}) of a $$thin plate of constant density $$\delta$$ covering the region between $$y = \sec^2$ x$$, $$-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$$ and the $$x-$$axis.

Accepted Answer

$$ \bar{x} = 0  by $$symmetry, $$ \bar{y} = \frac{\int_{-\pi/4}^{\pi/4} \frac{1}{2} \sec^4$ x dx}{\int_{-\pi/4}^{\pi/4} \sec^2$ x dx} $$

Question 11

Use the shell method to find the volume of the solid generated by revolving the region bounded by $$ x = y^2$ $$, $$ x = -y $$, $$ y = 2 $$, and $$ y \geq 0 $$ about the $$x-$$axis.

Accepted Answer

$$ V = 2\pi \$$int_0$$^2 y (y^2$ + y) dy = 2\pi \$$int_0$$^2 (y^3$ + y^2$) dy $$

Question 12

Evaluate $$ \int \sec^2$(3x + 2) dx .$$

Accepted Answer

$$ \frac{1}{3} 	an(3x + 2) + C $$

Question 13

A bathroom scale is compressed $$\frac{1}{16} in by a 150 lb $$person. Assuming Hooke's Law, how much does someone who compresses the scale $$\frac{1}{8} in $$weigh?

Accepted Answer

$$ 300 	ext{ lb} $$

Question 14

A bathroom scale is compressed $$\frac{1}{8} in. $$How much work is done compressing the scale this far, if a 150 lb person compresses it $$\frac{1}{16} in$$? (Assume Hooke's Law.)

Accepted Answer

$$ W = \frac{1}{2} k x^2$ = \frac{1}{2} \cdot 2400 \cdot \left( \frac{1}{8} \$$right)^2$$ = 18.75 	ext{ in-lb} $$

Question 15

Find the volume of the solid generated by revolving the region bounded by the $$y-$$axis, the line $$y = 1$$, and the curve $$x = 	an\left( \frac{\pi}{4} y ight)$$ about the $$y-$$axis.

Accepted Answer

$$ V = \pi \$$int_0$$^1 	an^2$\left( \frac{\pi}{4} y ight) dy $$

Question 16

Evaluate $$ \$$int_0$$^1 \frac{x^3$}{\sqrt{x^4$ + 9}} dx .$$

Accepted Answer

$$ \frac{1}{6} (\sqrt{10} - 3) $$

Work Done Lifting a Bucket and Rope (Work as an Integral)

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