The graph of your lemniscate looks like the symbol for infinity.
The equation has no solutions where is negative; the two loops are formed from the two intervals where is positive.
integrate f(x,y)=1/((1+x^2+y^2))^2 over the region enclosed by one loop of the leminisciate r^2=cos2(theta)
here i know it has to be converted into polar coordinates
what would be the limits
as i dont know the grapho of r^2=cos2(theta)