The function I have is

with

where

I want to integrate over this region. So I noticed its symmetric first of all. So my plan is to only consider the positive half first and then double the resulting area I find. Secondly I find that I can separate this region into 2 regions. (See attached image)

Region 1: A triangle with base = height = .

Region 2: upper limit of y is arc created by and lower limit is . And x ranges from 0 to .

I find when I go to do the double integral for region 2 I keep getting a negative answer.

The integral for region 2 I'm tyring is:

Any suggestions? Am I on the right track? Or completely off?

Thanks in advance.