Someone Please Have Mercy and Help Me Set Up This Triple Integral

[TMO]

Homework Statement

Evaluate ∫∫∫_E x² dV where E is the solid that lies within the cylinder x² + y² = 1, above the plane z = 0, and below the cone z² = 4x² + 4y². Solve it using cylindrical coordinates.

Homework Equations

dxdydz = dzdrdθ, and other typical anti-derivative tricks from Calculus II.

The Attempt at a Solution

I have been attempting to solve this abomination for three straight hours and I am still nowhere. Instead of showing all of my attempts, I will show one such:

We know that the radius must remain between zero and one, since it cannot exceed the encompassing cylinder. We also know that there are no restrictions on its rotation. Lastly we know that x² = r² cos² θ and that r² = x² + y². Thus z = 2r. Using this, we may set up the integral as following:

∫₀² ∫₀^2π ∫₀^z/2 r³ cos² θ drdθdz

 

[K^2]

I'll make your life a lot simpler. Based on symmetry, what can you tell me about ∫∫∫_E x² dV vs ∫∫∫_E y² dV? Now, wouldn't computing ∫∫∫_E r² dV be a lot easier? You should now have just two regions to consider, each with a simple cylindrical integral and no trig functions.