Looks to me it's:
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
Hmm. The only difference between your integral and the one I had was that I had as a factor, but you had . So I tracked the ps in this manner:
becomes ;
becomes ;
and the jacobian contributes .
. Have I made an error?
Using the integral you've provided, I evaluated it fully out, leaving the phi integral for last, and obtained:
After using a u-substitution of u = cos(phi), I have the integral
Which then evaluates to 0. I guess the answer really is zero...