Exam 8: Further Applications of Integration

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 lb/ft³.) (ft)

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

Write an integral giving the area of the surface obtained by revolving the curve about the x-axis. (Do not evaluate the integral.) y = on [3, 6]

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. m₁ = 3, m₂ = 4, m₃ = 5 P₁ (-3, 5), P₂ (3, 4), P₃ (-4, 1)

For any normal distribution, find to two decimal places.

Find the area of the surface obtained by revolving the graph of y = on [0, 1] about the x-axis.

Find the centroid of the region bounded by the given curves.

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 lb/ft³.) (ft)

Write an integral giving the arc length of the graph of the equation from P to Q. y = ; P (-2, ), Q (5, )

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes. The manager wants to advertise that anybody who isn't served within a certain number of minutes gets a free hamburger. But she doesn't want to give away free hamburgers to more than % of her customers. What value of x must she use in the advertisement "if you aren't served within x minutes, you get a free hamburger"?

A gate in an irrigation canal is constructed in the form of a trapezoid ft wide at the bottom, ft wide at the top, and 2 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..

Find the centroid of the region bounded by the graphs of the given equations.

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.)

Find the length of the curve for the interval .

Find the centroid of the region bounded by the given curves.

Let the function whose graph is shown be a probability density function. Calculate the mean.

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 minutes. Find the median waiting time.

Find the area of the region under the graph of f on [a, b].

Find the arc length of the graph of the given equation on the specified interval. y = ( + 1) , [2, 5]

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. = x - 3 ; [0, 3]