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. $y = e ^ { x } , 1 \leq y \leq 9 ; y \text {-axis }$

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 $62.5 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

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 $10 .$

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 $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

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 from $\mathbf { A }$ to $\mathbf { B }$ .

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. $y = x ^ { 3 } , x + y = 2 , x = 0$

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.

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 $v = 245$

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