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 y-axis. -y = , y = 0, x = 1, x = 16

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 x = 3 , x = - 3y, and y = 1 about the line y = 1

Solve the problem. -A diving pool that is 10 m deep and full of water has a viewing window on one of its vertical walls. Find the force on a square window, 2.5 m on a side, with the lower edge of the window 1.5 m from the bottom of the pool.

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region in the first quadrant bounded by x = 3y - and the y-axis about the y-axis

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 = 4x, y = ; revolve about the y-axis

Write the integral that gives the surface area generated when the curve is revolved about the x-axis. Do not simplify. -y = ln on [1, ]

Solve the problem. -The following figure shows the shape and dimensions of a small dam. Assuming the water level is at the top of the dam, find the total force on the face of the dam. Round to one decimal place when appropriate.

Set up an integral for the length of the curve. -x = + 6y, 0 $\le$ y $\le$ 6

Solve the problem. -A plate shaped like an equilateral triangle 1 m on a side is placed on a vertical wall 2 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.

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 = 5 , y = 0, x = 0, x = 1

Find the area of the shaded region. -

Find the area enclosed by the given curves. -y = x³, y = 4x

Solve the problem. -Find a curve through the point (0, 4) whose length integral, 0 $\le$ x $\le$ 1, is L =

Set up an integral for the length of the curve. - + 6y = 6x - 1, 1 $\le$ y $\le$ 2

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

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 = ln on [1, ]

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by x = , x = 0, y = - 2, y = 2

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 = , y = 3 + 2x, for x $\ge$ 0

Find the volume of the described solid. -The base of a solid is the region between the curve y = 4cos x and the x-axis from x = 0 to . The cross sections perpendicular to the x-axis are squares with bases running from the x-axis to the curve.

Find the length of the curve. -y = 2 from x = 0 to x =