Volumes-by-Cross-Sections Review KEY

 

The following questions refer to the first-quadrant region bounded by  and .

 

Set-up but do not evaluate the integrals.

 

 

 

 

 

 

Helpful notes for all 10 problems:

, , , , functions intersect at  and

 

All of the rotation problems can be done with washers OR shells.  Both answers are given and either is acceptable.

 

 

 

         1.          Find the volume of the region generated when rotated about the x-axis.

            washers: , shells:

         2.          Find the volume of the region generated when rotated about the y-axis.

            washers: , shells:

         3.          Find the volume of the region generated when rotated about y = -2.

            washers: , shells:

         4.          Find the volume when cross-sections perpendicular to the x-axis are sides of squares.

           

         5.          Find the volume of the region generated when rotated about x = 2.

            washers: , shells:

         6.          Find the volume when cross-sections perpendicular to the y-axis are legs of isosceles right triangles.

           

         7.          Find the volume of the region generated when rotated about y = 5.

            washers: , shells:

         8.          Find the volume when cross-sections perpendicular to the x-axis are diameters of circles.

           

         9.          Find the volume of the region generated when rotated about x = -3.

            washers: , shells:

     10.          Find the volume when cross-sections perpendicular to the y-axis are sides of squares.