Exam 8: Further Applications of Integration

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 centroid of the region shown in the figure.

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 $P ( - 1 < X < 1 )$ .

The masses $m _ { i }$ are located at the points $P _ { i }$ . Find the moments $M _ { x }$ and $M _ { y }$ and the center of mass of the system. =5,=9,=10 (1,5),(3,-2),(-2,-1)

The marginal cost function $C ^ { \prime } ( x )$ is defined to be the derivative of the cost function. If the marginal cost of manufacturing $x$ units of a product is $C ^ { \prime } ( x ) = 0.009 x ^ { 2 } - 1.8 x + 9$ (measured in dollars per unit) and the fixed start-up cost is $C ( 0 ) = 2,200,000$ , use the Total Change Theorem to find the cost of producing the first 5,000 units.

The marginal revenue from producing $x$ units of a certain product is $100 + x - 0.001 x ^ { 2 } + 0.00003 x ^ { 3 }$ (in dollars per unit). Find the increase in revenue if the production level is raised from 1,100 units to 1,700 units.

Find the centroid of the region bounded by the given curves. $y = 6 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }$

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 16 - x ^ { 2 } , y = 16$ , and $x = 4$ about the $y$ -axis.

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

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 .

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.

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

Write an integral giving the area of the surface obtained by revolving the curve about the $x$ -axis. (Do not evaluate the integral.) $y = \frac { 2 } { x } \text { on } [ 1,5 ]$

Find the area of the surface obtained by rotating the circle $x ^ { 2 } + y ^ { 2 } = 7 ^ { 2 }$ about the line $y = 7$ .

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

Find the centroid of the region bounded by the graphs of the given equations. $y = 10 - x ^ { 2 } , \quad y = 2 - 2 x$

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. a. $13.86$ minutes b. $17.86$ minutes c. $15.86$ minutes d. $16.86$ minutes e. $14.86$ minutes

Write an integral giving the area of the surface obtained by revolving the curve about the $x$ -axis. (Do not evaluate the integral.) $y = \frac { 2 } { x } \text { on } [ 4,5 ]$

Find the centroid of the region bounded by the graphs of the given equations. $y = | x | \sqrt { 16 - x ^ { 2 } } , \quad y = 0 , \quad x = - 4 , \quad x = 4$