Triple Integrals in Spherical Coordinates:
1. Evaluate \( \displaystyle{ \int_{-1}^1
\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}}
\: \: e^{(x^2+y^2+z^2)^{3/2}} \,\, dz \, dy\, dx} \)
solution
2. Evaluate \(\displaystyle{ \int_{-3}^3
\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}}
\sqrt{x^2+y^2+z^2} \,\, dz \, dy\, dx} \)
solution
3. Evaluate \( \displaystyle{ \iiint_B (x^2+y^2+z^2) dV}\), where
\(B\) is the portion of the unit ball in the first octant.
solution
4. Rewrite the following (do not evaluate) triple integral in
spherical coordinates \( \displaystyle{ \iiint_E y^2 \, dV}\), where \(E\)
is the solid hemisphere \(x^2+y^2+z^2 \le 9, \: y \ge 0 \).
solution
5. Convert \(\displaystyle{ \int_0^3
\int_0^{\sqrt{9-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{18-x^2-y^2}}
(x^2+y^2+z^2) \, dz\,dx\,dy}\) into spherical coordinates.
solution
6. Rewrite the following (do not evaluate) triple integral in
spherical coordinates \(\displaystyle{ \iiint_S (x+y+z) \, dV} \)
where \( S\) is the solid region above the cone \(z=\sqrt{3x^2+3y^2}\)
and inside the sphere \(x^2+y^2+z^2=16\).
solution