Question

Can you guys help me to solve the double integral of squr( x^2 - Y^2) dx dy where x^2 - Y^2 >=0 and 0<=x<=1?

Answer 1

Draw your domain of integration. x^2-y^2 = 0 describes a pair of intersecting lines, y=x and y=-x (or a degenerate hyperbola if you want to call it that). To take care of the inequality, just plug in some points. You have four regions in the xy-plane, and you can find that the inequality is satisfied in the left and right regions. Now you apply your restriction 0<=x<=1, and your domain of integration is a triangle.

You can describe this triangular region by 0<x<1 and -x<y<x (here I am using < to really mean <=), and so should integrate with respect to y first, then with respect to x.

Answer 2

If you want to check your work, you can use Wolfram Alpha online.
http://www.wolframalpha.com/
It will calculate double and triple integrals for you. One syntax that works is
integrate (function) dy dx, x=a..b, y=c..d
(replace "function" and "a,b,c,d" with the values you want.)