Question

Use spherical coordinates to evaluate the triple integral ∭E​y²dV, where E is the solid hemisphere defined by x²+y²+z² ≤ 9 and z ≥ 0.

Answer 1

To evaluate the triple integral ∭E​y²dV using spherical coordinates, we need to express the element of volume dV in terms of spherical coordinates.

The limits for the given solid hemisphere E are r: 0 to 3, θ: 0 to π/2, φ: 0 to 2π. By substituting these limits and the expression for dV into the integral, we can simplify and evaluate to obtain the result.

Step-by-step explanation:

To evaluate the triple integral ∭E​y²dV using spherical coordinates, we first need to express the element of volume dV in terms of spherical coordinates. In spherical coordinates, dV = r²sinθ dr dθ dφ, where r is the radial coordinate, θ is the polar angle, and φ is the azimuthal angle.

For the given solid hemisphere E, we have the following limits: r: 0 to 3, θ: 0 to π/2, φ: 0 to 2π. Substituting these limits and the expression for dV into the integral, we get:

∭E​y²dV = ∫02π ∫0π/2 ∫03 (rsinθ)²(r²sinθ dr) dθ dφ

Simplifying and evaluating this triple integral will give us the desired result.