Math 2AChapter 12 In-class Test Solutions

1.      Find the volume and surface area of the solid that the cylinder  cuts out of the sphere of radius a centered at the origin.
SOLN:  The volume can be found by

The surface area of the spherical caps of the cylindrical cutout can be computed by parameterizing the surface as, say, 

The surface area of the cylindrical sides can be computed using the parameterization .  Thus 

Since where the cylinder and the sphere intersect we have  and substituting leads to .  The surface area is .
Thus the total surface area is

2.      (a)  Evaluate , where n is an integer and D is the region bounded by
       the circles with center the origin and radii r and R, .


          SOL’N:  Substitute  so that


(b)  For what values of n does the integral in part (a) have a limit as  ?
       
             SOL’N:     if and only if n < 2.

(c)  Find , where E is the region bounded by the spheres with      
      center the origin and radius r and R, .  


           SOL’N:  
                 


(d)  For what values of n does the integral in part (c) have a limit as  ?

         SOL’N:  For this limit to exist we need n < 3.