Exam 8: Further Applications of Integration

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 µ = $4000$ . What is the median lifetime of these lightbulbs? Give your answer rounded to two decimal places.

A trough is filled with a liquid of density 855 kg/ $\mathrm{m}^{3}$ . The ends of the trough are equilateral triangles with sides 9 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

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 curves. $y=5 \ln 2 x, y=0, x=\frac{e}{\pi}$

Find the centroid of the region shown in the figure.

Find the arc length function for the curve $y=10 x^{3 / 2}$ with starting point $P_{0}(1,15)$ .

Set up, but do not evaluate, an integral for the length of the curve. $y=x^{3} \sqrt{5-x}, 0 \leq x \leq 2$

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

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

Find the arc length of the graph from A to B.

Write an integral giving the arc length of the graph of the equation from P to Q. y = $\frac{1}{x^{2}+9}$ ; P (-2, $\frac{1}{13}$ ), Q (5, $\frac{1}{34}$ )

A hawk flying at an altitude of $121$ m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation $y=121-\frac{x^{2}}{64}$ until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. True or False? The distance traveled by the prey from the time it is dropped until the time it hits the ground is approximately $156.532$ m.

Set up, but do not evaluate, an integral for the length of the curve. $y=3 x+\sin x, 0 \leq x \leq 5 \pi$

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-axis

An aquarium is 4 ft long, 3 ft wide, and 2 ft deep. If the aquarium is filled with water, find the force exerted by the water (a) on the bottom of the aquarium, (b) on the longer side of the aquarium, and (c) on the shorter side of the aquarium. (The weight density of water is 62.4 lb/ft³.)

A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitoes is increasing at an estimated rate of $2,100+7 e^{0.7 t}$ per week (where t is measured in weeks). By how much does the mosquito population increase between the 4th and 6 th weeks of summer?

Find the length of the curve. $x=\frac{y^{4}}{8}+\frac{1}{4 y^{2}}, 1 \leq y \leq 3$

Find the area of the surface obtained by rotating the curve about the y-axis. $y=\frac{1}{4} x^{2}-\frac{1}{2} \ln x, 1 \leq x \leq 8$

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.

A steady wind blows a kite due west. The kite's height above ground from horizontal position $x=0$ to $x=50$ ft is given by $y=100-\frac{1}{50}(x-20)^{2}$ . Find the distance traveled by the kite. Give your answer rounded to two decimal places.