where E is the solid that lies between the cylinders
and
,
above the xy-plane, and below the plane z = x+2. Then, using cylindrical
coordinates, evaluate the integral.
Solution: The boundary E is sketched below in blue. It is defined by the annular cylinder, intersected by the plane z = x+ 2.
As for the evaluation of the integral
note two things: First, as the graph above shows, the volume elements of E are
symmetric to the y-axis. Second, the integrand, y, is positive when y >0
and negative when y <0; the positive and negative balance out for this
integral. Thus the value of the integral is 0.