3R) Please help!!

Question

Use cylindrical coordinates to evaluate

∫∫∫_E (√x²+y² ) (dV),

where E is the region inside the cylinder (x − 1)² + y² = 1 and between the planes z = −1 and z = 1.

Answer 1

Let r = sqrt(x²+y²)

The (x-1)² + y² = 1 can be expressed in r,θ: (using identity and x = rcosθ, y=rsinθ)

r²cos²θ -2rcosθ +r²sin²θ + 1 -1 = 0 or r = 2cosθ (movement of a unit 1 circle to the right 1 unit)

We want to integrate z, r, then θ

Triple integral(z from -1 to 1, r from 0 to 2cosθ, and θ from 0 to 2π) of r (rdzdrdθ) (last part is dV)

Integral (r from 0 to 2cosθ, and θ from 0 to 2π) of 2r (rdrdθ)

Integral of (θ from 0 to 2pi) of (2/3 r³) evaluated from 0 to 2cosθ (dθ)

integral from 0 to 2π of (16/3)cos³ θ dθ

substitute (1-sin²θ)cosθθ for integrand and integrate.