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## Homework Statement

Find the volume of the solid bounded above by the surface z = x^2 + y^2 and below by the

triangular region in the xy-plane enclosed by the lines x = 0 , y = x , and x + y = 8.

## Homework Equations

V = ∫∫ Height

Base

## The Attempt at a Solution

I first found height, because height = z (upper) - z (lower) = (x^2 + y^2) - 0 = x^2 + y^2

Afterwards, I began solving for the base. I know the base is enclosed by the curves y = x,

y = x - 8, and x = 0. I proceed assuming the region is y-simple, giving me the boundaries:

Base { x ≤ y ≤ 8-x

and 0 ≤ x ≤ 4 }

This sets up the double integral for me, which turns out to be:

4 8-x

∫ { ∫ (x^2 + y^2) dy } dx

0 x

Solving this integral gives me that volume is 1024/3.

Could someone look through my work and see if I made any errors? I'm a beginner at setting up these equations, so I suspect my integral to have a mistake. Thank you in advance.