Mathematics - Definite Integration Question with Solution | TestHub

Given,

${\int }_0^2\left(\sqrt{2x}-\sqrt{2x-x^2}\right)dx=$

${\int }_0^1\left(1-\sqrt{1-y^2}-\frac{y^2}{2}\right)dy+{\int }_1^2\left(2-\frac{y^2}{2}\right)dy+I$

Solving LHS $={\int }_0^2\left(\sqrt{2x}-\sqrt{2x-x^2}\right)dx=\frac{8}{3}-\frac{\pi }{2}$

Now, solving RHS $={\int }_0^1\left(1-\sqrt{1-y^2}-\frac{y^2}{2}\right)dy+{\int }_1^2\left(2-\frac{y^2}{2}\right)dy+I$

$=I+\frac{5}{3}-\frac{\pi }{4}$

Now equating LHS and RHS we get,  $I=1-\frac{\pi }{4}={\int }_0^1\left(1-\sqrt{1-y^2}\right)dy$ as ${\int }_0^1\left(\sqrt{1-y^2}\right)dy=\frac{\mathrm{\pi }}{4}$