JEE Mains · Maths · STD 12 - 7.2 definite integral
\(\text { If } \int\limits_{0}^{2}\left(\sqrt{2 x}-\sqrt{2 x-x^{2}}\right) d x=\) \(\int\limits_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right) d y+\int\limits_{1}^{2}\left(2-\frac{y^{2}}{2}\right) d y+I\) then \(I=\dots\dots\dots\)
- 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 & Solution
Correct Answer
(C) \(\int\limits_{0}^{1}\left(1-\sqrt{1-y^{2}}\right) d y\)
Step-by-step Solution
Detailed explanation
\(LHS =\int\limits_{0}^{2}\left(\sqrt{2 x }-\sqrt{2 x - x ^{2}}\right) dx =\frac{8}{3}-\frac{\pi}{2}\) \(RHS =\int\limits_{0}^{1}\left(1-\sqrt{1- y ^{2}}-\frac{ y ^{2}}{2}\right) dy +\int\limits_{1}^{2}\left(2-\frac{ y ^{2}}{2}\right) dy + I\) \(I +\frac{5}{3}-\frac{\pi}{4}\)…
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