Find area of region bound by
y^2=x^2-x^4
First try and draw a graph of this curve.
By symmetry the area is 4 times the area between the curve and the x-axis from x = 0 to x = 1 of $\displaystyle y = \sqrt{x^2 - x^4} = x \sqrt{1 - x^2}$.
To do the integral I suggest making the substitution $\displaystyle u = 1 - x^2$.