Deck 6: Applications of Integration

Question 1

Find the area bounded by the curves $y=x^{2}-2 x$ and $y=3$ .

A) $\frac{20}{3}=6 \frac{2}{3}$
B) $\frac{32}{3}=10 \frac{2}{3}$
C) 10
D) 9

Answer 1

$\frac{32}{3}=10 \frac{2}{3}$

Question 2

Find the area bounded by the curves $y=2 x$ and $y^{2}=4 x$ .

A) $\frac{1}{6}$
B) $\frac{1}{3}$
C) $\frac{4}{3}=1 \frac{1}{3}$
D) 1

Answer 2

$\frac{1}{3}$

Question 3

Find the volume of the solid formed by revolving the region under $y=5 x^{2}$ from $x=1$ to $x=3$ about the $\mathrm{x}$ -axis.

A) 242
B) 1210
C) $1210 \pi$
D) $242 \pi$

Answer 3

$1210 \pi$

Question 4

Find the volume of the solid formed by revolving the region under $y=5 x^{2}$ from $x=1$ to $x=3$ about the $y$ -axis.

A) $100 \pi$
B) $200 \pi$
C) $150 \pi$
D) $250 \pi$

Question 5

Find the center of mass of the two- dimensional system which has a mass of 3 at (4, 2), a mass of 2 at $(6,10)$ , and a mass of 7 at $(0,6)$ .

A) $(3,5)$
B) $\left[2, \frac{17}{3}\right]$
C) $\left[\frac{31}{12}, \frac{17}{3}\right]$
D) $\left[\frac{31}{12}, \frac{17}{12}\right]$

Question 6

Find the centroid of the plane region in the first quadrant bounded above by $y=9$ and below by $y=$ $\mathrm{x}^{2}$ .

A) $\left[\frac{9}{8}, \frac{27}{5}\right]=(1.125,5.4)$
B) $(1,5)$
C) $\left[\frac{23}{18}, \frac{97}{18}\right]$
D) $\left[\frac{25}{18}, \frac{97}{18}\right]$

Question 7

Find the average value of the function $f(x)=\frac{1}{x^{2}}$ over the interval from $x=1$ to $x=5$ .

A) $\frac{4}{5}$
B) $\frac{\ln 25}{4}$
C) $\frac{1}{4}$
D) $\frac{1}{5}$

Question 8

Find the amount of work done in winding all of a 300 - $\mathrm{ft}$ hanging cable that weighs $120 \mathrm{lb}$ .

A) $24,000 \mathrm{ft}-\mathrm{lb}$
B) $18,000 \mathrm{ft}-\mathrm{lb}$
C) $36,000 \mathrm{ft}-\mathrm{lb}$
D) $15,000 \mathrm{ft}-\mathrm{lb}$

Question 9

For the problems below, find each area bounded by the curves.

- $y=2x^{2}-5x-8$ y=x+12

Question 10

For the problems below, find each area bounded by the curves.

- $y=x^{3}$ $y=2 x$

Question 11

Use the disk method to find the volume of the solid formed by revolving the region bounded by $y=0.5 x^{2}, x=0$ , and $y=6$ about the $y$ - axis.

Question 12

Use the shell method to find the volume of the solid formed by revolving the region bounded by $\mathrm{y}=\sqrt{\mathrm{x}}, \mathrm{y}=0$ , and $\mathrm{x}=9$ about the $\mathrm{x}$ - axis.

Question 13

Use any method to find the volume of the solid formed by revolving the region bounded by $y=x^{2}$ and $y=3 x$ in the first quadrant about the $y$ - axis.

Question 14

There is a mass of 14 at (- 10,0 $)$ and a mass of 26 at ( 6,0 ). Find where mass of 12 should be placed on the $x$ - axis so that $(5,0)$ is the center of mass.

Question 15

Find the center of mass for the two- dimensional system: $m_{1}=8$ at $(3,5), m_{2}=5$ at $(-1,-3), m_{3}=20$ at $(-2,4)$ .

Question 16

Find the center of mass measured from the lighter end of a straight wire that is $16 \mathrm{~cm}$ long and whose density is given by $\mathrm{Q}(\mathrm{x})=8+\mathrm{x}^{2}$ , where $\mathrm{x}$ is the distance from one end.

Question 17

Find the centroid of the region bounded by $y=9-x^{2}$ and $y=0$ .

Question 18

Find the moment of inertia and the radius of gyration of the region bounded by $y=12-x^{2}$ , $\mathrm{y}=3$ , and $\mathrm{x}=0$ about the $\mathrm{y}$ - axis. $\mathrm{Q}=4$

Question 19

Find the moment of inertia and the radius of gyration of the solid formed by revolving the region bounded by $\mathrm{y}=\mathrm{x}^{2}, \mathrm{y}=0$ , and $\mathrm{x}=4$ about the $\mathrm{x}$ - axis. $\mathrm{\varrho}=5$

Question 20

A vertical gate in a dam is in the shape of an isosceles trapezoid $16.0 \mathrm{ft}$ across the top and $10.0 \mathrm{ft}$ across the bottom, with a height of $12.0 \mathrm{ft}$ . Find the force against the gate if the water surface is at the top of the gate.