Exam 6: Applications of Integration

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -x = 3 , x = - 3y, y = 3

Find the volume of the described solid. -The solid lies between planes perpendicular to the x-axis at x = - 2 and x = 2. The cross sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = to the parabola y = 8 - .

Use a calculator to approximate the area of the surface generated when the given curve is revolved about the x-axis. Round to two decimal places when necessary. -y = sinx on

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -y = 2x, y = 4x, y = 2

Find the length of the curve. -x = from y = 4 to y = 9

Find the volume of the described solid. -The solid lies between planes perpendicular to the x-axis at x = - 3 and x = 3. The cross sections perpendicular to the x-axis are semicircles whose diameters run from to

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -y = 7 , y = 7

Find the length of the curve. -x = - from x = 9 to x = 16

Find the area of the surface generated when the given curve is revolved about the x-axis. -y = on

Solve the problem. -Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 20 cos 5t, v(0) = 8, s(0) = 12

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -x = 6 , x = 6

Solve the problem. -A water tank is formed by revolving the curve y = 6 about the y-axis. Find the volume of water in the tank as a function of the water depth, y.

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. -y = 5x, y = 0, x = 4; revolve about the x-axis

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by x = , x = 0, y = 27

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the x-axis

Find the area of the surface generated when the given curve is revolved about the x-axis. -y = + on

Solve the problem. -A water trough has a semicircular cross section with a radius of 0.5 m and a length of 4 m (see figure). How much work is required to pump water out of the trough when it is full? Round to two decimal places when appropriate.

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the x-axis

Solve the problem. -A square plate 3 m on a side is placed on a vertical wall 3 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then compute the force on each plate. Round to three decimal places when appropriate.

Find the length of the curve. -y = , 0 $\le$ x $\le$