0-a-x-a-2-x-2-a-2-x-2-

Question Number 151894 by peter frank last updated on 24/Aug/21

\int_{0}^{a} x \sqrt{\frac{a^{2} - x^{2}}{a^{2} + x^{2}}}

Commented by peter frank last updated on 24/Aug/21

limit from a \to 0

Answered by Ar Brandon last updated on 24/Aug/21

I = \int_{0}^{a} x \sqrt{\frac{a^{2} - x^{2}}{a^{2} + x^{2}}} d x = \frac{1}{2} \int_{0}^{a^{2}} \sqrt{\frac{a^{2} - u}{a^{2} + u}} d u

= \frac{a^{2}}{2} \int_{0}^{1} \sqrt{\frac{1 - v}{1 + v}} d v = \frac{a^{2}}{2} \int_{0}^{1} \frac{1 - v}{\sqrt{1 - v^{2}}} d v

= \frac{a^{2}}{2} {[\arcsin (v) + \sqrt{1 - v^{2}}]}_{0}^{1} = \frac{a^{2}}{2} (\frac{π}{2} - 1)

- a ⩽ x ⩽ a

Commented by peter frank last updated on 24/Aug/21

more clarification please

step one and two

Commented by Ar Brandon last updated on 24/Aug/21

Change of variable u = x^{2} = a^{2} v