Question 1

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.

Accepted Answer

A)    100π100 \pi100π 
B)    200π200 \pi200π 
C)    150π150 \pi150π 
D)    250π250 \pi250π B

Question 2

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

Accepted Answer

A)    16\frac{1}{6}61​ 
B)    13\frac{1}{3}31​ 
C)    43=113\frac{4}{3}=1 \frac{1}{3}34​=131​ 
D)   1  B

Question 3

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 $(0,6)$.

Accepted Answer

A)    (3,5) (3,5) (3,5)  
B)    [2,173]\left[2, \frac{17}{3}\right][2,317​] 
C)    [3112,173]\left[\frac{31}{12}, \frac{17}{3}\right][1231​,317​] 
D)    [3112,1712]\left[\frac{31}{12}, \frac{17}{12}\right][1231​,1217​] B

Question 4

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 $(-2,4)$.

Accepted Answer

The answer of Find the center of mass for the...

Question 5

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.

Accepted Answer

The answer of Use the shell method to find the...

Question 6

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.

Accepted Answer

The answer of Use any method to find the volume...

Question 7

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

Accepted Answer

A)    203=623\frac{20}{3}=6 \frac{2}{3}320​=632​ 
B)    323=1023\frac{32}{3}=10 \frac{2}{3}332​=1032​ 
C)   10
D)   9

Question 8

For the problems below, find each area bounded by the curves.
-$y=x^{3}$ $y=2 x$

Accepted Answer

The answer of For the problems below, find each area...

Question 9

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.

Accepted Answer

The answer of Use the disk method to find the...

Question 10

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

Accepted Answer

A)    24,000ft−lb24,000 \mathrm{ft}-\mathrm{lb}24,000ft−lb 
B)    18,000ft−lb18,000 \mathrm{ft}-\mathrm{lb}18,000ft−lb 
C)    36,000ft−lb36,000 \mathrm{ft}-\mathrm{lb}36,000ft−lb 
D)    15,000ft−lb15,000 \mathrm{ft}-\mathrm{lb}15,000ft−lb

Question 11

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

Accepted Answer

A)    [98,275]=(1.125,5.4) \left[\frac{9}{8}, \frac{27}{5}\right]=(1.125,5.4) [89​,527​]=(1.125,5.4)  
B)    (1,5) (1,5) (1,5)  
C)    [2318,9718]\left[\frac{23}{18}, \frac{97}{18}\right][1823​,1897​] 
D)    [2518,9718]\left[\frac{25}{18}, \frac{97}{18}\right][1825​,1897​]

Question 12

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

Accepted Answer

A)    45\frac{4}{5}54​ 
B)    ln⁡254\frac{\ln 25}{4}4ln25​ 
C)    14\frac{1}{4}41​ 
D)    15\frac{1}{5}51​

Question 13

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

Accepted Answer

The answer of Find the centroid of the region bounded...

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.

Accepted Answer

The answer of There is a mass of 14 at...

Question 15

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.

Accepted Answer

A)   242
B)   1210
C)    1210π1210 \pi1210π 
D)    242π242 \pi242π

Question 16

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. $\mathrm{\varrho}=5$

Accepted Answer

The answer of Find the moment of inertia and the...

Question 17

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.

Accepted Answer

The answer of A vertical gate in a dam is...

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. $\mathrm{Q}=4$

Accepted Answer

The answer of Find the moment of inertia and the...

Question 19

For the problems below, find each area bounded by the curves.
-$y=2x^{2}-5x-8$ y=x+12

Accepted Answer

The answer of For the problems below, find each area...

Question 20

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.

Accepted Answer

The answer of Find the center of mass measured from...

Exam 6: Applications of Integration

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.

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

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 $(0,6)$ .

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 $(-2,4)$ .

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.

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.

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

For the problems below, find each area bounded by the curves. - $y=x^{3}$ $y=2 x$

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.

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

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

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

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

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.

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.

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. $\mathrm{\varrho}=5$

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.

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. $\mathrm{Q}=4$

For the problems below, find each area bounded by the curves. - $y=2x^{2}-5x-8$ y=x+12

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.

Exam 6: Applications of Integration

Find the volume of the solid formed by revolving the region under y=5x2y=5 x^{2}y=5x2 from x=1x=1x=1 to x=3x=3x=3 about the yyy -axis.

Find the area bounded by the curves y=2xy=2 xy=2x and y2=4xy^{2}=4 xy2=4x .

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) (6,10) (6,10) , and a mass of 7 at (0,6) (0,6) (0,6) .

Find the center of mass for the two- dimensional system: m1=8m_{1}=8m1​=8 at (3,5),m2=5(3,5), m_{2}=5(3,5),m2​=5 at (−1,−3),m3=20(-1,-3), m_{3}=20(−1,−3),m3​=20 at (−2,4)(-2,4)(−2,4) .

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

Use any method to find the volume of the solid formed by revolving the region bounded by y=x2y=x^{2}y=x2 and y=3xy=3 xy=3x in the first quadrant about the yyy - axis.

Find the area bounded by the curves y=x2−2xy=x^{2}-2 xy=x2−2x and y=3y=3y=3 .

For the problems below, find each area bounded by the curves. - y=x3y=x^{3}y=x3 y=2xy=2 xy=2x

Use the disk method to find the volume of the solid formed by revolving the region bounded by y=0.5x2,x=0y=0.5 x^{2}, x=0y=0.5x2,x=0 , and y=6y=6y=6 about the yyy - axis.

Find the amount of work done in winding all of a 300 - ft\mathrm{ft}ft hanging cable that weighs 120lb120 \mathrm{lb}120lb .

Find the centroid of the plane region in the first quadrant bounded above by y=9y=9y=9 and below by y=y=y= x2\mathrm{x}^{2}x2 .

Find the average value of the function f(x) =1x2f(x) =\frac{1}{x^{2}}f(x) =x21​ over the interval from x=1x=1x=1 to x=5x=5x=5 .

Find the centroid of the region bounded by y=9−x2y=9-x^{2}y=9−x2 and y=0y=0y=0 .

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 xxx - axis so that (5,0)(5,0)(5,0) is the center of mass.

Find the volume of the solid formed by revolving the region under y=5x2y=5 x^{2}y=5x2 from x=1x=1x=1 to x=3x=3x=3 about the x\mathrm{x}x -axis.

Find the moment of inertia and the radius of gyration of the solid formed by revolving the region bounded by y=x2,y=0\mathrm{y}=\mathrm{x}^{2}, \mathrm{y}=0y=x2,y=0 , and x=4\mathrm{x}=4x=4 about the x\mathrm{x}x - axis. ϱ=5\mathrm{\varrho}=5ϱ=5

A vertical gate in a dam is in the shape of an isosceles trapezoid 16.0ft16.0 \mathrm{ft}16.0ft across the top and 10.0ft10.0 \mathrm{ft}10.0ft across the bottom, with a height of 12.0ft12.0 \mathrm{ft}12.0ft . Find the force against the gate if the water surface is at the top of the gate.

Find the moment of inertia and the radius of gyration of the region bounded by y=12−x2y=12-x^{2}y=12−x2 , y=3\mathrm{y}=3y=3 , and x=0\mathrm{x}=0x=0 about the y\mathrm{y}y - axis. Q=4\mathrm{Q}=4Q=4

For the problems below, find each area bounded by the curves. - y=2x2−5x−8y=2x^{2}-5x-8y=2x2−5x−8 y=x+12

Find the center of mass measured from the lighter end of a straight wire that is 16 cm16 \mathrm{~cm}16 cm long and whose density is given by Q(x)=8+x2\mathrm{Q}(\mathrm{x})=8+\mathrm{x}^{2}Q(x)=8+x2 , where x\mathrm{x}x is the distance from one end.

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.

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

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 $(0,6)$ .

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 $(-2,4)$ .

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.

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.

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

For the problems below, find each area bounded by the curves. - $y=x^{3}$ $y=2 x$

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.

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

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

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

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

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.

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.

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. $\mathrm{\varrho}=5$

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.

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. $\mathrm{Q}=4$

For the problems below, find each area bounded by the curves. - $y=2x^{2}-5x-8$ y=x+12

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.