Question

Use spherical coordinates to evaluate ∫∫∫_E z dz dϕ dθ, where E is the region bounded by the surfaces x^2 + y^2 + z^2 = 1.

(a) (1/4) π
(b) (1/3) π
(c) (1/3) π
(d) (2/3) π

Answer 1

In spherical coordinates, integrating (z) over the region (E) bounded by
`\(x^2 + y^2 + z^2 = 1\)` results in the given expression. The evaluation of the integral leads to the correct option (b) ((1/3) π).

Explanation:

In spherical coordinates, the given triple integral ∫∫∫_E z dz dϕ dθ represents the integration of the variable (z) over the region (E), which is defined by the surface (x^2 + y^2 + z^2 = 1). The spherical coordinates involve the variables (θ) and (ϕ) representing the azimuthal and polar angles, respectively. The limits of integration for (θ) are [0, 2π], for (ϕ) are [0, π], and for (z) are [-1, 1], as the surface equation implies a unit sphere.

Upon performing the integration with respect to (z) first, the expression simplifies to ∫_0^(2π) ∫_0^π [(1/2)z^2]_{-1}^1 dϕ dθ. Evaluating this integral yields the final result of ((1/3) π), making option (b) the correct choice.

This solution involves successive integrations and application of the limits of integration based on the spherical coordinates, resulting in the determination of the volume integral over the specified region (E). The correct evaluation confirms that the integral over the region bounded by the unit sphere leads to the answer of ((1/3) π).