# Math Help - Triple Integral Problem Using Spherical Coordinates

1. ## Triple Integral Problem Using Spherical Coordinates

Given that:

Integral[-a, 0] Integral[-Sqrt(a^2-y^2), 0] Integral[a-Sqrt(a^2-x^2-y^2), a] dz dx dy

http://tinypic.com/view.php?pic=4ibucl&s=6

Please write an equivalent integral in spherical coordinates.

2. bump

3. Hi Dynas, Looks to me your integral:

$\int_{-a}^0 \int_{-\sqrt{a^2-y^2}}^0 \int_{a-\sqrt{a^2-x^2-y^2}}^a dz dx dy$

is the integral from the bottom of the sphere surface in the plot below, to the plane z=a shown in blue. Now, just because the integrand $f(x,y,z)=1$, that's the same as just the volume of 1/8 of a sphere of radius $a$ in spherical coordinates:

$\int_{0}^{\pi/2}\int_0^{\pi/2} \int_0^a \rho^2 \sin(\phi) d\rho d\theta d\phi=(1/8) 4/3 \pi a^3$

4. Thanks a lot. That was a big help.