Question

Express the integral as an iterated integral in six different ways, where E is the solid bounded by √(x^2 + y^2) ≤ z ≤ 1. Note: You can earn partial credit on this.

∫ from 0 to 2π ∫ from 0 to 1 ∫ from r to 1 r dz dr dθ
∫ from 0 to 1 ∫ from 0 to 2π ∫ from r to √(1 - z^2) r dθ dz dr

Answer 1

Six different ways to express the integral as an iterated integral were provided, considering the cylindrical symmetry of the volume bounded by a cone and a plane in cylindrical coordinates.

Step-by-step explanation:

To express the integral as an iterated integral in six different ways, considering the solid E bounded by √(x^2 + y^2) ≤ z ≤ 1, we need to take into account the cylindrical symmetry of the solid. The original integrals are already in cylindrical coordinates (r, θ, z), which are natural for this problem since the bounds for z depend on r. Below are six different ways to express the integral over this domain:
∫ from 0 to 2π ∫ from 0 to √(1 - z^2) ∫ from z to 1 r dr dθ dr
∫ from 0 to 1 ∫ from r to √(1 - z^2) ∫ from 0 to 2π r dθ dz dr

Each expression considers a different order of integration, but all cover the same volume E. The choice of which order to use will depend on convenience and the integrand's nature.