Exam 6: Applications of Integration

Set up an integral for the length of the curve. -y = 5 cot x, $\le$ x $\le$

Find the area of the shaded region. -

Solve the problem. -An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve - 5 $\le$ x $\le$ 5, about the x-axis (dimensions are in feet). How many cubic feet of fuel will the tank hold to the nearest cubic foot?

Solve the problem. -A glass has circular cross sections that taper (linearly) from a radius of 7 cm at the top of the glass to a radius of 6 cm at the bottom. The glass is 15 cm high and full of lemonade. How much work is required to drink all the lemonade through a straw if your mouth is 8 cm above the top of the glass? Assume the density of lemonade equals the density of water. Round to two decimal places when appropriate.

Solve the problem. -It takes 100 J of work to stretch a spring 1 m from its equilibrium position. How much work is needed to stretch it an additional 1 m?

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 = 8 , y = 8

Solve the problem. -Hooke's law is applicable to idealized (linear) springs that are not stretched or compressed too far. Consider a nonlinear spring whose restoring force is given by F(x) = 15x - 0.3 , for |x| $\le$ 8. How much work is done in compressing the spring from its equilibrium position (x = 0) to x = -0.5? Round to two 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 about the given lines. - , y = 9, x = 3; revolve about the line Y = 9

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

Write the integral that gives the surface area generated when the curve is revolved about the x-axis. Do not simplify. -y = tanx on

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 = 5y - and the y-axis about the x-axis

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

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the -y = , y = 1, x = 0 to x =

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

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 = 4x, y = 8x, x = 4

Solve the problem. -A water tank is formed by revolving the curve y = 7 about the y-axis. Water drains from the tank through a small hole in the bottom of the tank. At what constant rate does the water level, y, fall? (Use Torricelli's Law: dV/dt = -m .)

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.

Find the volume of the solid generated by revolving the shaded region about the given 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 = , y = 7, x = 0

Provide an appropriate response. -Which of the following integrals, if any, calculates the area of the shaded region?