Evaluate:

Triple integral over domain H of

To the point: I used spherical coordinates and during the integration with respect to phi (the vertical angle), it resulted in 0, causing the entire triple integral to evaluate to 0, which I assume to be wrong.

Elaborate: Using...

x = p cos(theta) sin(phi)

y = p sin(phi) sin(theta)

z = p cos(phi)

p =

and H = { (p, theta, phi) | 0 <= p <= 1, 0 <= theta <= pi,0 <= phi <= pi}

First I'll evaluate the integral with respect to phi:

integral(0, pi, p^6 * cos^3(phi) * sin(phi), dphi)

Using u = cos(phi) as the substitution, the integral becomes:

integral(1, -1, - u^3 / 7, du)

Which evalutes to zero.

Could anyone show me what I'm doing wrong, or confirm that zero is the answer?

Thank you!

David