Exam 8: Further Applications of Integration

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 mg/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 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. $y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]$

If $f ( x )$ is the probability density function for the blood cholesterol level of men over the age of 40 , where $x$ is measured in milligrams per deciliter, express as an integral the probability that the cholesterol level of such a man lies between 195 and $245 .$

A gate in an irrigation canal is constructed in the form of a trapezoid $4 \mathrm { ft }$ wide at the bottom, $8 \mathrm { ft }$ wide at the top, and $2 \mathrm { ft }$ high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate.

Use Simpson's Rule with $n = 6$ to estimate the length of the curve $y = 3 \sin x , 0 \leq x \leq \pi$ . Round your answer to six decimal places.

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

A type of lightbulb is labeled as having an average lifetime of 4000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean $\mu = 4000$ . What is the median lifetime of these lightbulbs? Give your answer rounded to two decimal places.

Find the centroid of the region bounded by the graphs of the given equations. $y = x ^ { 2 / 3 } , y = 0 , \quad x = 8$

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

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.

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

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 movie theater has been charging $\$ 10.00$ per person and selling about 500 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every $\$ 0.5$ that they lower the price, the number of moviegoers will increase by 50 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at $\$ 8$ .

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

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

Find the center of mass of the lamina of the region shown if the density of the circular lamina is six times that of the rectangular lamina.

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

A rectangular tank has width $3 \mathrm { ft }$ , height $2 \mathrm { ft }$ , and length $7 \mathrm { ft }$ . It is filled with equal volumes of water and oil. The oil has a weight density of $50 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ and floats on the water. Find the force exerted by the mixture on one end of the tank. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .)

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 $8 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.