1 Attachment(s)

Double Integral - Weird Region

The function I have is

$\displaystyle

f(x, y) = c(y^2 - x^2) e^{-y}

$

with $\displaystyle |x| \leq y \leq (\frac{1}{4} - x^2)\ and -\alpha \leq x \leq \alpha $

where $\displaystyle \alpha \approx 0.207 $

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 = $\displaystyle \alpha $.

Region 2: upper limit of y is arc created by $\displaystyle 1/4 - x^2 $ and lower limit is $\displaystyle \alpha $. And x ranges from 0 to $\displaystyle \alpha $.

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:

$\displaystyle

\int_0^\alpha \! \int_\alpha^{\frac{1}{4} - x^2} f(x, y)\,dy\,dx.\

$

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

Thanks in advance.