Question

Evaluate ${\int }_0^a\sqrt{a^2-x^2}dx$

Answer 1

${\int }_0^a\sqrt{a^2-x^2}dx$
Let $x=asiny$
$dx=acosydy$
${\int }_0^a\sqrt{a-x^2}dx$
$={\int }_0^{\frac{\pi }{2}}\sqrt{a^2-a^{2}si{n}^{2}y}.acosydy$
$I={\int }_0^{\frac{\pi }{2}}{a}^{2}co{s}^{2}ydy$
Applying properties of definite integrals
$I={\int }_0^{\frac{\pi }{2}}{a}^{2}co{s}^2(\frac{\pi }{2}-y)dy={\int }_0^{\frac{\pi }{2}}{a}^{2}si{n}^{2}ydy$
Therefore
$2I={\int }_0^{\frac{\pi }{2}}{a}^2(co{s}^{2}y+si{n}^{2}y).dy$
$2I={\int }_0^{\frac{\pi }{2}}{a}^{2}dy$
$2I=\frac{a^2\pi }{2}$
$I=\frac{a^2\pi }{4}$.