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Question Number 123598 by aurpeyz last updated on 26/Nov/20

p r o v e t h a t \underset{0}{\int^{a}} \sqrt{a^{2} - x^{2}} d x = \frac{π a^{2}}{4}

Answered by Dwaipayan Shikari last updated on 26/Nov/20

\int_{0}^{a} \sqrt{a^{2} - x^{2}} d x x = a s i n θ

\int_{0}^{\frac{π}{2}} a c o s θ \sqrt{a^{2} - a^{2} {s i n}^{2} θ} d θ

= a^{2} \int_{0}^{\frac{π}{2}} {c o s}^{2} θ = \frac{a^{2}}{2} \int_{0}^{\frac{π}{2}} 1 + c o s 2 θ d x

= \frac{π a^{2}}{4} + \frac{a^{2}}{2} \int_{0}^{\frac{π}{2}} c o s 2 θ = \frac{π a^{2}}{4} + \frac{a^{2}}{4} {[c o s 2 θ]}_{0}^{\frac{π}{2}} = \frac{π a^{2}}{4}