Use spherical coordinates the evaluate the following triple

Use spherical coordinates the evaluate the following triple integral:

Solution

Another more direct approach using

f(x)f(x)ndx=1n+1f(x)n+1+C:f(x)f(x)ndx=1n+1f(x)n+1+C:

0101xyx2+y2dydx=01xdx12012yx2+y2dy=1201xdx23(x2+y2)3/210=0101xyx2+y2dydx=01xdx12012yx2+y2dy=1201xdx23(x2+y2)3/2|01=

=1301x((x2+1)3/2x3)dx=16012x(x2+1)3/2dx1301x4dx==1301x((x2+1)3/2x3)dx=16012x(x2+1)3/2dx1301x4dx=

=115(x2+1)5/210115=115(4211)=215(221)