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MathCalculus

Evaluate the Integral by Changing to Spherical Coordinates

Convert a first-octant triple integral bounded by a cone and a sphere to spherical coordinates, then evaluate the transformed integral.

Question

Evaluate the integral by changing to spherical coordinates:

I=∫03∫09−y2∫x2+y218−x2−y2(x2+y2+z2) dz dx dy.I=\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.

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Evidence boundary

The Cartesian integral is taken from the cited Texas A&M course notes. The geometric interpretation, spherical bounds, Jacobian and exact value are shown as reproducible calculations. The answer uses the standard convention that φ is measured down from the positive z-axis.