Question 1

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis.
$$y = e ^ { x } , 1 \leq y \leq 9 ; y \text {-axis }$$

Accepted Answer

The answer of Set up, but do not evaluate, an...

Question 2

A swimming pool is $10 \mathrm { ft }$ wide and $36 \mathrm { ft }$ long and its bottom is an inclined plane, the shallow end having a depth of $4 \mathrm { ft }$ and the deep end, $12 \mathrm { ft }$. If the pool is full of water, find the hydrostatic force on the shallow end. (Use the fact that water weighs $62.5 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of A swimming pool is \(10 \mathrm {...

Question 3

The standard deviation for a random variable with probability density function $f$ and mean $\mu$ is defined
$$\sigma = \left[ \int _ { - \infty } ^ { \infty } ( x - \mu ) ^ { 2 } f ( x ) d x \right] ^ { 1 / 2 }$$
Find the standard deviation for an exponential density function with mean $10 .$

Accepted Answer

A)  10 
B)  2 
C)  5 
D)  $3.3$ 
E)  $2.5$

Question 4

A cylindrical drum of diameter $6 \mathrm { ft }$ and length $9 \mathrm { ft }$ is lying on its side, submerged in water $10 \mathrm { ft }$ deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of A cylindrical drum of diameter \(6 \mathrm...

Question 5

Find the area of the surface obtained by revolving the given curve about the $x$-axis.
$y = \frac { e ^ { x } + e ^ { - x } } { 2 }$ on $[ 0 , \ln 3 ]$

Accepted Answer

A)  $\pi \ln 3$ 
B)  $\pi \left( \frac { 20 } { 9 } + \ln 3 \right)$ 
C)  $\frac { 20 } { 9 } \pi$ 
D)  $\frac { \pi } { 3 } \left( \frac { 20 } { 9 } + \ln 3 \right)$

Question 6

Find the area of the surface obtained by revolving the given curve about the $x$-axis.
$$y = \frac { e ^ { x } + e ^ { - x } } { 2 } \text { on } [ 0 , \ln 5 ]$$

Accepted Answer

The answer of Find the area of the surface obtained...

Question 7

Set up, but do not evaluate, an integral for the length of the curve.
$$y = x \sqrt [ 3 ] { 5 - x } , 0 \leq x \leq 9$$

Accepted Answer

The answer of Set up, but do not evaluate, an...

Question 8

Let $f ( x ) = \frac { 2 c } { \left( 1 + x ^ { 2 } \right) }$
a) For what value of $c$ is $f$ a probability density function?
b) For that value of $c$, find $P ( - 1 < X < 1 )$.

Accepted Answer

The answer of Let \(f ( x ) = \frac...

Question 9

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$. The ends of the trough are equilateral triangles with sides $6 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Accepted Answer

The answer of A trough is filled with a liquid...

Question 10

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is 20 minutes. Find the median waiting time.

Accepted Answer

A)  $13.86$ minutes 
B)  $17.86$ minutes 
C)  $15.86$ minutes 
D)  $16.86$ minutes 
E)  $14.86$ minutes

Question 11

Find the centroid of the region bounded by the graphs of the given equations.
$$y = 9 - x ^ { 2 } , \quad y = 3 - x$$

Accepted Answer

A)  $\left( \frac { 5 } { 2 } , \frac { 1 } { 2 } \right)$ 
B)  $\left( \frac { 1 } { 2 } , \frac { 5 } { 2 } \right)$ 
C)  $\left( 5 , \frac { 1 } { 2 } \right)$ 
D)  $\left( \frac { 1 } { 2 } , 5 \right)$

Question 12

Find the area of the surface obtained by rotating the curve about the $y$-axis. Select the correct answer.
$$y = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \ln x , 1 \leq x \leq 4$$

Accepted Answer

A)  $\frac { 3 \pi } { 2 }$ 
B)  $\frac { 8 \pi } { 3 }$ 
C)  $\frac { 10 } { 3 }$ 
D)  $\frac { 11 \pi } { 2 }$ 
E)  None of these

Question 13

Find the arc length of the graph from $\mathbf { A }$ to $\mathbf { B }$.

Accepted Answer

A)  $\frac { 1 } { 54 } ( \sqrt { 10 } - 13 \sqrt { 13 } )$ 
B) 
$\frac { 1 } { 27 } ( \sqrt { 10 } - 13 \sqrt { 13 } )$ 
C) 
$\frac { 1 } { 27 } ( 80 \sqrt { 10 } - 13 \sqrt { 13 } )$ 
D)  $\frac { 1 } { 54 } ( 80 \sqrt { 10 } - 13 \sqrt { 13 } )$

Question 14

Find the arc length of the graph of the given equation on the specified interval.
$$y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,5 ]$$

Accepted Answer

The answer of Find the arc length of the graph...

Question 15

Find the centroid of the region bounded by the given curves.
$$y = x ^ { 3 } , x + y = 2 , x = 0$$

Accepted Answer

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

Question 16

For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given the demand curve
$$p = 60 - \frac { x } { 20 }$$
and the supply curve
$$p = 30 + \frac { x } { 30 } ,$$
find the consumer surplus.

Accepted Answer

The answer of For a given commodity and pure competition,...

Question 17

Find the coordinates of the centroid for the region bounded by the curves $y = 4 x ^ { 2 } , x = 0$, and $y = 144$.

Accepted Answer

The answer of Find the coordinates of the centroid for...

Question 18

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$. The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the
Select the correct answer.

Accepted Answer

A)  $3.5 \times 10 ^ { 5 } \mathrm {~N}$ 
B)  $4.5 \times 10 ^ { 5 } \mathrm {~N}$ 
C)  $6.5 \times 10 ^ { 5 } \mathrm {~N}$ 
D)  $2.5 \times 10 ^ { 5 } \mathrm {~N}$ 
E)  $5.5 \times 10 ^ { 5 } \mathrm {~N}$

Question 19

Find the coordinates of the centroid for the region bounded by the curves $y = 5 x ^ { 2 } , x = 0$, and $v = 245$

Accepted Answer

The answer of Find the coordinates of the centroid for...

Question 20

$$\text { Find the center of mass of a lamina in the shape of a quarter-circle with radius } 15 \text { with density } \rho = 3 \text {. }$$

Accepted Answer

The answer of \[\text { Find the center of mass...

Exam 8: Further Applications of Integration

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis. $y = e ^ { x } , 1 \leq y \leq 9 ; y \text {-axis }$

The standard deviation for a random variable with probability density function $f$ and mean $\mu$ is defined $\sigma = \left[ \int _ { - \infty } ^ { \infty } ( x - \mu ) ^ { 2 } f ( x ) d x \right] ^ { 1 / 2 }$ Find the standard deviation for an exponential density function with mean $10 .$

Find the area of the surface obtained by revolving the given curve about the $x$ -axis. $y = \frac { e ^ { x } + e ^ { - x } } { 2 }$ on $[ 0 , \ln 3 ]$

Find the area of the surface obtained by revolving the given curve about the $x$ -axis. $y = \frac { e ^ { x } + e ^ { - x } } { 2 } \text { on } [ 0 , \ln 5 ]$

Set up, but do not evaluate, an integral for the length of the curve. $y = x \sqrt [ 3 ] { 5 - x } , 0 \leq x \leq 9$

Let $f ( x ) = \frac { 2 c } { \left( 1 + x ^ { 2 } \right) }$ a) For what value of $c$ is $f$ a probability density function? b) For that value of $c$ , find $P ( - 1 < X < 1 )$ .

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ . The ends of the trough are equilateral triangles with sides $6 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is 20 minutes. Find the median waiting time.

Find the centroid of the region bounded by the graphs of the given equations. $y = 9 - x ^ { 2 } , \quad y = 3 - x$

Find the area of the surface obtained by rotating the curve about the $y$ -axis. Select the correct answer. $y = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \ln x , 1 \leq x \leq 4$

Find the arc length of the graph of the given equation on the specified interval. $y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,5 ]$

Find the centroid of the region bounded by the given curves. $y = x ^ { 3 } , x + y = 2 , x = 0$

Find the coordinates of the centroid for the region bounded by the curves $y = 4 x ^ { 2 } , x = 0$ , and $y = 144$ .

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ . The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the Select the correct answer.

Find the coordinates of the centroid for the region bounded by the curves $y = 5 x ^ { 2 } , x = 0$ , and $v = 245$

$\text { Find the center of mass of a lamina in the shape of a quarter-circle with radius } 15 \text { with density } \rho = 3 \text {. }$ $\text { Find the center of mass of a lamina in the shape of a quarter-circle with radius } 15 \text { with density } \rho = 3 \text {. }$

Exam 8: Further Applications of Integration

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis. y=ex,1≤y≤9;y-axis y = e ^ { x } , 1 \leq y \leq 9 ; y \text {-axis }y=ex,1≤y≤9;y-axis

Find the area of the surface obtained by revolving the given curve about the xxx -axis. y=ex+e−x2y = \frac { e ^ { x } + e ^ { - x } } { 2 }y=2ex+e−x​ on [0,ln⁡3][ 0 , \ln 3 ][0,ln3]

Find the area of the surface obtained by revolving the given curve about the xxx -axis. y=ex+e−x2 on [0,ln⁡5]y = \frac { e ^ { x } + e ^ { - x } } { 2 } \text { on } [ 0 , \ln 5 ]y=2ex+e−x​ on [0,ln5]

Set up, but do not evaluate, an integral for the length of the curve. y=x5−x3,0≤x≤9y = x \sqrt [ 3 ] { 5 - x } , 0 \leq x \leq 9y=x35−x​,0≤x≤9

Let f(x)=2c(1+x2)f ( x ) = \frac { 2 c } { \left( 1 + x ^ { 2 } \right) }f(x)=(1+x2)2c​ a) For what value of ccc is fff a probability density function? b) For that value of ccc , find P(−1<X<1)P ( - 1 < X < 1 )P(−1<X<1) .

A trough is filled with a liquid of density 855 kg/m3855 \mathrm {~kg} / \mathrm { m } ^ { 3 }855 kg/m3 . The ends of the trough are equilateral triangles with sides 6 m6 \mathrm {~m}6 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is 20 minutes. Find the median waiting time.

Find the centroid of the region bounded by the graphs of the given equations. y=9−x2,y=3−xy = 9 - x ^ { 2 } , \quad y = 3 - xy=9−x2,y=3−x

Find the area of the surface obtained by rotating the curve about the yyy -axis. Select the correct answer. y=14x2−12ln⁡x,1≤x≤4y = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \ln x , 1 \leq x \leq 4y=41​x2−21​lnx,1≤x≤4

Find the arc length of the graph from A\mathbf { A }A to B\mathbf { B }B .

Find the arc length of the graph of the given equation on the specified interval. y=23(x2+1)3/2,[2,5]y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,5 ]y=32​(x2+1)3/2,[2,5]

Find the centroid of the region bounded by the given curves. y=x3,x+y=2,x=0y = x ^ { 3 } , x + y = 2 , x = 0y=x3,x+y=2,x=0

Find the coordinates of the centroid for the region bounded by the curves y=4x2,x=0y = 4 x ^ { 2 } , x = 0y=4x2,x=0 , and y=144y = 144y=144 .

A trough is filled with a liquid of density 855 kg/m3855 \mathrm {~kg} / \mathrm { m } ^ { 3 }855 kg/m3 . The ends of the trough are equilateral triangles with sides 7 m7 \mathrm {~m}7 m long and vertex at the bottom. Find the hydrostatic force on one end of the Select the correct answer.

Find the coordinates of the centroid for the region bounded by the curves y=5x2,x=0y = 5 x ^ { 2 } , x = 0y=5x2,x=0 , and v=245v = 245v=245

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis. $y = e ^ { x } , 1 \leq y \leq 9 ; y \text {-axis }$

Find the area of the surface obtained by revolving the given curve about the $x$ -axis. $y = \frac { e ^ { x } + e ^ { - x } } { 2 }$ on $[ 0 , \ln 3 ]$

Find the area of the surface obtained by revolving the given curve about the $x$ -axis. $y = \frac { e ^ { x } + e ^ { - x } } { 2 } \text { on } [ 0 , \ln 5 ]$

Set up, but do not evaluate, an integral for the length of the curve. $y = x \sqrt [ 3 ] { 5 - x } , 0 \leq x \leq 9$

Let $f ( x ) = \frac { 2 c } { \left( 1 + x ^ { 2 } \right) }$ a) For what value of $c$ is $f$ a probability density function? b) For that value of $c$ , find $P ( - 1 < X < 1 )$ .

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ . The ends of the trough are equilateral triangles with sides $6 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Find the centroid of the region bounded by the graphs of the given equations. $y = 9 - x ^ { 2 } , \quad y = 3 - x$

Find the area of the surface obtained by rotating the curve about the $y$ -axis. Select the correct answer. $y = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \ln x , 1 \leq x \leq 4$

Find the arc length of the graph of the given equation on the specified interval. $y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,5 ]$

Find the centroid of the region bounded by the given curves. $y = x ^ { 3 } , x + y = 2 , x = 0$

Find the coordinates of the centroid for the region bounded by the curves $y = 4 x ^ { 2 } , x = 0$ , and $y = 144$ .

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ . The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the Select the correct answer.

Find the coordinates of the centroid for the region bounded by the curves $y = 5 x ^ { 2 } , x = 0$ , and $v = 245$