Exam 7: Applications of Integration

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is feet and the gasoline weighs 42 pounds per cubic foot.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations , and about the line .

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region bounded by about the x-axis.

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. ​

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by and in the first quadrant about the -axis.

Find the area of the region bounded by equations by integrating (i) with respect to x and (ii) with respect to y.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Find the arc length of the graph of the function over the interval [1,216].

A porthole on a vertical side of a submarine (submerged in seawater) is square . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

If the accumulation function F(x) is given by , evaluate F(4)

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

Find the center of mass of the point masses lying on the x-axis.

Find the area of the region bounded by the graphs of the function . Round your answer to three decimal places.

Set up and evaluate integrals for finding the moment about the y-axis for the region bounded by the graphs of the equations. (Assume ρ =1.) .

Two electrons repel each other with a force that varies inversely as the square of the distance between them, where k is the constant of proportionality. One electron is fixed at the point . Find the work done in moving the second electron from to . ​ ​

Consider a beam of length L = 5 feet with a fulcrum x feet from one end as shown in the figure. In order to move a 550-pound object, a person weighing 184 pounds wants to balance it on the beam. Find x (the distance between the person and the fulcrum) such that the system is equilibrium. Round your answer to two decimal places.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line .

Use the disk or shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line .

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis.