Question

Give the limits of integration for evaluating the integral

∭f(r,θ,z)dz r dr dθ

as an iterated integral over the region that is bounded below by the plane z = 0, on the side by the cylinder r=cosθ and on top by the paraboloid z=3r^2.

Answer 1

θ = 0 to θ = 2π, r = 1 to r = -1 and z = 0 to z = 3

Explanation:

Since the integration is in cylindrical coordinates, θ is integrated from 0 to 2π.

Since r = cosθ

when θ = 0, r = cos0 = 1

when θ = 2π, r = cos2π = -1

Since the region is bounded below by the plane z = 0, and on the top by the paraboloid z = 3r², z is integrated from z = 0 to z = 3r².

So, when r = 1, z = 3r² = 3(1)² = 3

when r = -1, z = 3r² = 3(-1)² = 3

So, the limits of integration of

∭f(r,θ,z)dz r dr dθ are θ = 0 to θ = 2π, r = 1 to r = -1 and z = 0 to z = 3