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MathCalculus

Evaluate the Following Integral in Spherical Coordinates: A Unit Hemisphere

Evaluate OpenStax Example 5.47 directly in spherical coordinates and check the result as a unit-hemisphere volume.

Question

Evaluate the following integral in spherical coordinates:

I=∫02π∫0π/2∫01ρ2sin⁡ϕ dρ dϕ dθ.I=\int_0^{2\pi}\int_0^{\pi/2}\int_0^1 \rho^2\sin\phi\,d\rho\,d\phi\,d\theta.

The ranges are 0≤ρ≤10\le\rho\le1, 0≤ϕ≤π/20\le\phi\le\pi/2, and 0≤θ≤2π0\le\theta\le2\pi; ϕ\phi is measured down from the positive zz-axis.

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Evidence boundary

The result applies to the triple integral and spherical-coordinate convention in OpenStax Example 5.47. The source page prints the inner lower limit as p=0, but its own solution uses ρ=0\rho=0; this page follows that consistent notation. The task evaluates an integral already in spherical coordinates; it does not ask for a coordinate conversion.