Question

If $\int\limits_{0}^{2}\left(\sqrt{2 x}-\sqrt{2 x-x^{2}}\right) d x=$
$\int\limits_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right) d y+\int\limits_{1}^{2}\left(2-\frac{y^{2}}{2}\right) d y+I$

• A. $\int\limits_{0}^{1}\left(1+\sqrt{1- y ^{2}}\right) d y$
• B. $\int\limits_{0}^{1}\left(\frac{y^{2}}{2}-\sqrt{1-y^{2}}+1\right) d y$
• C. $\int\limits_{0}^{1}\left(1-\sqrt{1-y^{2}}\right) d y$
• D. $\int\limits_{0}^{1}\left(\frac{y^{2}}{2}+\sqrt{1-y^{2}}+1\right) d y$

Answer 1

Answer: (C)

$LHS =\int\limits_{0}^{2}\left(\sqrt{2 x}-\sqrt{2 x-x^{2}}\right) d x=\frac{8}{3}-\frac{\pi}{2}$
$R H S=\int\limits_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right) d y+\int\limits_{1}^{2}\left(2-\frac{y^{2}}{2}\right) d y+I$
$I+\frac{5}{3}-\frac{\pi}{4}$
So, $I=1-\frac{\pi}{4}=\int\limits_{0}^{1}\left(1-\sqrt{1-y^{2}}\right) d y$