Exam 6: Applications of Integration

Solve the problem. -In a certain memory experiment, subject A is able to memorize words at a rate given by In the same memory experiment, subject B is able to memorize at the rate given by How many more words does subject B memorize from t = 0 to t = 14 (during the first 14 minutes)?

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = + 4, y = 2x + 4

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the x-axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = , y = 36, x = 0

Set up an integral for the length of the curve. -x = , 0 $\le$ y $\le$ 2

Solve the problem. -A 60-m-long, 9.1-mm-diameter rope hangs free from a ledge. The density of the rope is 70 g/m. How much work is needed to pull the entire rope to the ledge?

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = , y = - x + 5

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 y-axis. -y = 5x, y = - , x = 1

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by y = 2 , y = 2, and x = 0 about the line x = 1

Solve the problem. -A cylindrical water tank has height 12 m and radius 2 m (see figure). If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank? Express the answer in terms of $\pi$ .

Solve the problem. -A spherical water tank with an inner radius of 4 m has its lowest point 3 m above the ground. It is filled by a pipe that feeds the tank at its lowest point (see figure). Neglecting the volume of the inflow pipe, how much work is required to fill the tank if it is initially empty? Express the answer in terms of $\pi$ .

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 y-axis. -y = 7 , y = 7x, for x $\ge$ 0

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 = 16, v(0) = 7, s(0) = 11

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = - 3x + 6, y = 3x, x = 0

Solve the problem. -A swimming pool is 16 m long and 10 m wide, with a bottom that slopes uniformly from a depth of at one end to a depth of 2 m at the other end (see figure). Assuming the pool is full, how much work is required to pump the water to a level 0.2 m above the top of the pool? Round to one decimal place when appropriate.

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis x =

Find the length of the curve. -x = , 1 $\le$ y $\le$ 9

Find the volume of the solid generated by revolving the region about the y-axis. -The region in the first quadrant bounded on the left by the circle + = 4, on the right by the line , and above by the line y = 2

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The triangle with vertices (0, 0), (0, 2), and ( 4, 2) about the line x = 4