Exam 7: Applications of Integration

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 lb/ft³.)

The base of S is the parabolic region Cross-sections perpendicular to the y axis are squares.Find the volume of S.

Find the volume of the solid obtained by rotating the region bounded by about the line

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)

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

Find the volume of the solid obtained by rotating the region bounded by about the x-axis.

True or False? The volume of a solid torus (the donut-shaped solid shown in the figure) with r = and R = is

Choose the differential equation corresponding to this direction field.

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of .

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of inhabitants, people have a disease at the beginning of the week and have it at the end of the week. How long does it take for of the population to be infected?

Find the area of the surface obtained by rotating the circle about the line .

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis. y-axis

Find the area of the surface obtained by revolving the given curve about the y-axis.x = on [0, 2]

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-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.)

Find the volume of the frustum of a pyramid with square base of side square top of side and height

Find the area of the region bounded by the parabola , the tangent line to this parabola at , and the x axis.

A steady wind blows a kite due west. The kite's height above ground from horizontal position to ft is given by .Find the distance traveled by the kite.Give your answer rounded to two decimal places.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.