Question

Find the limits of integration on the triple integral for the volume of the sphere using cartesian, cylindrical, and spherical coordinates and the function to be integrated. For your answers θ=θ, ϕ= phi, and rho= rho.

Answer 1

Cartesian Coordinates:

The limits for x, y, and z are from -r to r, where r is the radius of the sphere.

Cylindrical Coordinates:

The limits for ρ, θ, and z are ρ from 0 to r, θ from 0 to 2π, and z from -√(r^2 - ρ^2) to √(r^2 - ρ^2).

Spherical Coordinates:

The limits for ρ, θ, and ϕ are ρ from 0 to r, θ from 0 to 2π, and ϕ from 0 to π.

Step-by-step explanation:

In Cartesian coordinates, a sphere with a center at the origin and radius 'r' is defined by x^2 + y^2 + z^2 = r^2. For each coordinate (x, y, z), the limits are symmetrically from -r to r.

In cylindrical coordinates, for a sphere, the radius varies from 0 to the sphere's radius 'r' (ρ: 0 to r). The angle θ spans the entire circle (0 to 2π). The z-limits can be derived from the equation of the sphere in cylindrical coordinates: z = ±√(r^2 - ρ^2).

Spherical coordinates use ρ as the distance from the origin to a point, θ as the angle in the xy-plane, and ϕ as the angle from the positive z-axis. For a sphere, ρ ranges from 0 to the radius 'r', θ completes a full circle (0 to 2π), and ϕ covers the upper hemisphere (0 to π).

These limits ensure that the volume integral covers the entire volume of the sphere without redundancies, reflecting the spherical symmetry of the shape.