Exam 6: Applications of Integration

The base of a certain solid is a plane region R enclosed by the x-axis and the curve $y = 1 - x ^ { 2 }$ . Each cross-section of the solid perpendicular to the y-axis is an isosceles right triangle with hypotenuse lying in R. Find the volume of the solid.

Find the length of the curve $3 x = 2 y ^ { \frac { 3 } { 2 } } , ( 0,0 ) \text { to } \left( \frac { 2 } { 3 } , 1 \right)$ .

The graph of a continuous function g(x) is given below: List from smallest to largest: (a) The average value of g over $[ 0,10 ]$ (d) $\int _ { 3 } ^ { 5 } g ( x ) d x$ (b) The average rate of change of g over $[ 0,10 ]$ (e) $\int _ { 0 } ^ { 10 } g ( x ) d x$ (c) $g ^ { \prime } ( 6 )$ (f) $\int _ { 3 } ^ { 6 } g ( x ) d x$

Find the centroid of the region bounded by $y = 1 - x ^ { 2 } \text { and } y = 0$

The density function for the waiting time at a bank is modeled by $p ( t ) = 0.1 e ^ { - 0.1 t } , t > 0$ and is measured in minutes.(a) What is the probability that a customer will be served within the first 5 minutes? (b) What is the probability that a customer has to wait for more than 15 minutes?

A swimming pool 5 m wide, 10 m long, and 3 m deep is filled with seawater of density 1030 kg/m³ to a depth of 2.5 m. Find (a) the hydrostatic pressure at the bottom of the pool.(b) the hydrostatic force on the bottom.(c) the hydrostatic force on one end of the pool.

The demand function for producing a certain commodity is given by $p = 1000 - 0.1 x - 0.0001 x ^ { 2 }$ . Find the consumer surplus when the sale level is 500.

The demand function for a certain commodity is $p ( x ) = 4 - \frac { 1 } { 30 } x$ Find the consumer surplus when the sales level is 30.

Find the center of mass of the linear system $m _ { 1 } = 1 , m _ { 2 } = 9 , m _ { 3 } = 6$ $x _ { 1 } = - 3 , \quad x _ { 2 } = - 1$ $x _ { 3 } = 1$ .

Assume the weights of adult males are normally distributed with a mean weight of 150 pounds and a standard deviation of 20 pounds. Use Simpson's Rule or the Midpoint Rule to estimate the following: (a) What is the probability that an adult male chosen at random will weigh between 120 pounds and 180 pounds? (b) What percentage of the adult male population weighs more than 200 pounds?

Find the arc length of the curve $y = \sqrt { 4 - x ^ { 2 } } , - 2 \leq x \leq 2$ .

A rope 100 feet long weighing 2 pounds per foot hangs over the edge of a building 100 feet tall. How much work in foot-pounds is done in pulling the rope to the top of the building?

Let X be a continuous random variable with density function $p ( X ) = \left\{ \begin{array} { l l } \frac { 1 } { 9 } X ( 4 - X ) & \text { if } 0 \leq \mathrm { X } \leq 3 \\0 & \text { otherwise }\end{array} \right.$ What is the mean of X?:

A stone is dropped from a bell tower 100 feet tall. Find the average velocity of the stone from the instant it is dropped until it strikes the ground. (Assume that the acceleration due to gravity is 32 ft/s².)

The volume of the solid obtained by rotating the plane region enclosed by $y = x ^ { 4 } , y = 1 , \text { and } x = 0$ about the x = 1 is

City B (population 4,000) is 10 miles north of City A (population 6,000), and City C (population 5,000) is 30 miles east of City B. Where is the best place to locate a super market serving the people in these cities?

The following table shows the relationship between price and demand for milk produced in a large dairy. q (billions of pounds of milk per year) 45 50 55 60 65 70 75 p (price in dollars per pound) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 Determine the consumer's surplus when the sales quality is 65 billion pounds of milk in a year. Illustrate your answer by drawing the corresponding demand curve and the identifying the consumer's surplus as a region.

A spring stretches 1 foot beyond its natural position under a force of 100 points. How much work in foot-pounds is done in stretching it 3 feet beyond its natural position?

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

Find the area of the shaded region: