Question

Set up the triple integral for the volume of the sphere ϱ=4 in cylindrical coordinates. 8) A) ∫02π∫04∫016−r2dzdrdθ B) ∫02π∫04∫016−r2rdzdrdθ C) ∫02π∫04∫−16−r216−r2rdzdrdθ D) ∫02π∫04∫−16−r216−r2dzdrdθ Use cylindrical coordinates to find the volume of the indicated region

Answer 1

To set up the triple integral for the volume of the sphere ρ=4 in cylindrical coordinates, we express the sphere in terms of cylindrical coordinates and set the limits for each variable. The setup for the triple integral is (B) ∫0⁻2π∫0⁻4∫0⁻16-r^2(16-r^2)r dz dr dθ.

Step-by-step explanation:

To set up the triple integral for the volume of the sphere ρ=4 in cylindrical coordinates, we need to express the sphere in terms of cylindrical coordinates, which are (r, θ, z). The equation for the sphere is r=4. The limits for each variable are as follows: θ goes from 0 to 2π, z goes from 0 to 16-r^2, and r goes from 0 to 4.

The setup for the triple integral is as follows:

(B) ∫02π∫04∫016-r^2(16-r^2)r dz dr dθ