Show solutionHide solution+ Question 4 Find the exact area of the region enclosed by the curves y=x2−xandy=x.y = x\sqrt{2-x} \quad\text{and}\quad y = x. Question 4 - Solution The intersections satisfy x2−x=xx\sqrt{2-x}=x, giving x=0,1x=0,1. For 0<x<10<x<1, the square-root curve is above the line, hence A=∫01(x2−x−x)dx.A=\int_0^1\bigl(x\sqrt{2-x}-x\bigr)\,dx. With u=2−xu=2-x, ∫01x2−xdx=∫12(2−u)udu=[43u3/2−25u5/2]12=162−1415.\int_0^1x\sqrt{2-x}\,dx=\int_1^2(2-u)\sqrt u\,du =\left[\frac43u^{3/2}-\frac25u^{5/2}\right]_1^2 =\frac{16\sqrt2-14}{15}. A=322−4330>0.\boxed{A=\frac{32\sqrt2-43}{30}>0.}