Exam 8: Further Applications of Integration

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 10 minutes. Find the median waiting time.

Find the length of the curve. $y = \frac { 1 } { 6 } \left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } , 0 \leq x \leq 2$

Find the centroid of the region bounded by the graphs of the given equations. $y = 15 - x ^ { 2 } , \quad y = 3 - x$

A large tank is designed with ends in the shape of the region between the curves $y = \frac { x ^ { 2 } } { 2 }$ and $y = 15$ , measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of $9 \mathrm { ft }$ with gasoline. (Assume that the density of the gasoline is $42.0 \mathrm { lb } / \mathrm { tt } ^ { 3 }$ .)

A vertical plate is submerged in water (the surface of the water coincides with the $x$ -axis). Find the force exerted by the water on the plate. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

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

Dye dilution is a method of measuring cardiac output. If $A \mathrm { mg }$ of dye is used and $c ( t )$ is the concentration of the dye at time $t$ , then the cardiac output over the time interval $[ 0 , T ]$ is given by $F = \frac { A } { \int _ { 0 } ^ { T } c ( t ) d t }$ Find the cardiac output over the time interval $[ 0,15 ]$ if the dye dilution method is used with $10 \mathrm { mg }$ of dye and the dye concentration, in $\mathrm { mg } / \mathrm { L }$ , is modeled by $c ( t ) = \frac { 1 } { 2 } t ( 15 - t ) , 0 \leq t \leq 15$ where $t$ is measured in seconds.

A movie theater has been charging $\$ 9.00$ per person and selling about 300 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every $\$ 1$ that they lower the price, the number of moviegoers will increase by 40 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at $\$ 4$ .

A cylindrical drum of diameter $2 \mathrm { ft }$ and length $5 \mathrm { ft }$ is lying on its side, submerged in water $17 \mathrm { ft }$ deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

A vertical plate is submerged in water (the surface of the water coincides with the $x$ -axis). Find the force exerted by the water on the plate. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

Find the arc length of the graph of the given equation on the specified interval. $y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 3,4 ]$

A large tank is designed with ends in the shape of the region between the curves $y = \frac { x ^ { 2 } } { 2 }$ and $y = 15$ , measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of $9 \mathrm { ft }$ with gasoline. (Assume that the density of the gasoline is $42.0 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y = \ln x ^ { 8 }$ about the $x$ -axis on the interval $1 \leq x \leq 8$ .

Let $f ( x ) = \frac { 4 c } { \left( 1 + x ^ { 2 } \right) }$ . a) For what value of $c$ is $f$ a probability density function? b) For that value of $c$ , find $P ( - 1 < X < 1 )$ .

Find the centroid of the region bounded by the given curves. $y = 2 x ^ { 3 } , 2 x + y = 4 , x = 0$

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .) Select the correct answer.

Dye dilution is a method of measuring cardiac output. If $A \mathrm { mg }$ of dye is used and $c ( t )$ is the concentration of the dye at time $t$ , then the cardiac output over the time interval $[ 0 , T ]$ is given by $F = \frac { A } { \int _ { 0 } ^ { T } c ( t ) d t }$ Find the cardiac output over the time interval $[ 0,15 ]$ if the dye dilution method is used with $11 \mathrm { mg }$ of dye and the dye concentration, in $\mathrm { mg } / \mathrm { L }$ , is modeled by $c ( t ) = \frac { 1 } { 2 } t ( 15 - t ) , 0 \leq t \leq 15$ where $t$ is measured in seconds.

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

Find the area of the surface obtained by rotating the curve about the $x$ -axis. $x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } , 1 \leq y \leq 2$