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Question: If b \> a, and I = \(\int_{a}^{b}\sqrt{\frac{x - a}{b - x}}\)dx, then I equals...

If b > a, and I = abxabx\int_{a}^{b}\sqrt{\frac{x - a}{b - x}}dx, then I equals

A

π2\frac{\pi}{2} (b – a)

B

p (b – a)

C

p/2

D

2p (b – a)

Answer

π2\frac{\pi}{2} (b – a)

Explanation

Solution

Put b –x = t2, so that

I = ba0bt2at2\int_{\sqrt{b - a}}^{0}\sqrt{\frac{b - t^{2} - a}{t^{2}}} (–2t) dt

= 20cc2t2\int_{0}^{c}\sqrt{c^{2} - t^{2}}dt where c = ba\sqrt{b - a}

= 2 [12tc2+t2+c22sin1(tc)]0c\left\lbrack \frac{1}{2}t\sqrt{c^{2} + t^{2}} + \frac{c^{2}}{2}\sin^{- 1}\left( \frac{t}{c} \right) \right\rbrack_{0}^{c}

= 0 + c2 sin–1 (1) – 0

= π2\frac{\pi}{2} (b –a)