# Evaluate the Integral over the given region. BOUNDARIES

• Aug 1st 2010, 05:18 PM
boquist
Evaluate the Integral over the given region. BOUNDARIES
Here is a picture of the problem:
Attachment 18401

I know that the y values are bounded between y=-1 and y=1+x^2.
I am slightly confused about how the x values are bounded. They are bounded on the left by x=-1 and on the right by x=y^2, but how do I also show that it is bounded by x=1?
Do I need to evaluate the area by using three integrals?

Thank you for the help!
• Aug 1st 2010, 06:56 PM
lvleph
I would use three integrals
$\int_{-1}^{1 + x^2}\!\!\!\int_{-1}^0\! xy\,dx\,dy + \int_{\sqrt{x}}^{1+x^2}\!\!\!\int_{0}^1\! xy\,d\,xdy + \int_{-1}^{-\sqrt{x}}\!\!\!\int_{0}^1\! xy\,dx\,dy$
• Aug 2nd 2010, 01:50 AM
HallsofIvy
Here's another way to do it: The integral over the entire square is [tex]\int_{x= -1}^1\int_{y= -1}^2 xydy[tex].

Subtract from that the integral above the parabola $y= x^2+ 1$ and below y= 2, $\int_{x= -1}^1\int_{y= x^2+ 1}^2 xy dydx$, and the integral from $x= y^2$ to x= 1, $\int_{y= =1}^1\int_{x= y^2}^1 xydxdy$.
• Aug 2nd 2010, 08:22 AM
boquist
Perfect! Thanks to you both, its so simple I'm disappointed I never thought of doing it like that!