Final answer:
To evaluate the given integral, use spherical coordinates and set up the integral in terms of r, θ, and φ. Then perform the integrals in the specified order and calculate the final result.
Step-by-step explanation:
To evaluate the integral ∫_W dV, where W is one-eighth of a sphere with a radius of 3, we can use spherical coordinates.
The spherical coordinates are usually denoted by (ρ, θ, φ), where:
ρ is the radial distance from the origin,
θ is the polar angle (measured from the positive z-axis),
φ is the azimuthal angle (measured from the positive x-axis in the xy-plane).
The volume element in spherical coordinates is given by ρ²sinφ dρ dθ dφ.
For the one-eighth sphere with a radius of 3, we have the following limits for integration:
ρ: 0 to 3 (radius of the sphere),
θ: 0 to π/2 (one-eighth of the sphere),
φ: 0 to π/2 (one-eighth of the sphere).
Now, the integral becomes:
∫_W dV = ∫ from 0 to π/2 ∫ from 0 to π/2 ∫ from 0 to 3 ρ²sinφ dρ dθ dφ
Let's evaluate this integral step by step:
∫ from 0 to π/2 ∫ from 0 to π/2 ∫ from 0 to 3 ρ²sinφ dρ dθ dφ
= ∫ from 0 to π/2 ∫ from 0 to π/2 [(1/3)ρ³sinφ] evaluated from 0 to 3 dθ dφ
= ∫ from 0 to π/2 ∫ from 0 to π/2 [(1/3)(3)³sinφ - (1/3)(0)³sinφ] dθ dφ
= ∫ from 0 to π/2 ∫ from 0 to π/2 3²sinφ dθ dφ
= ∫ from 0 to π/2 [3²θ] evaluated from 0 to π/2 dφ
= ∫ from 0 to π/2 (9π/2) dφ
= (9π/2)φ evaluated from 0 to π/2
= (9π/2)(π/2 - 0)
= (9π²)/4
Therefore, the value of the integral ∫_W dV for the given one-eighth sphere with a radius of 3 is (9π²)/4.