Use the substitution $\displaystyle u = \arcsin (x/a) $to find the integral of $\displaystyle \sqrt(a^2-x^2)$ between x = 0 and x = b.
I've never confronted an integral as hard as this. Could someone please please show me how to do this???
Use the substitution $\displaystyle u = \arcsin (x/a) $to find the integral of $\displaystyle \sqrt(a^2-x^2)$ between x = 0 and x = b.
I've never confronted an integral as hard as this. Could someone please please show me how to do this???
$\displaystyle \sin{u} = \frac{x}{a}$
$\displaystyle x = a\sin{u}$
$\displaystyle dx = a\cos{u} \, du$
$\displaystyle \int \sqrt{a^2 - x^2} \, dx$
$\displaystyle \int \sqrt{a^2 - a^2\sin^2{u}} \cdot a\cos{u} \, du$
$\displaystyle \int \sqrt{a^2(1 - \sin^2{u})} \cdot a\cos{u} \, du$
$\displaystyle a^2 \int \cos^2{u} \, du$
use the double angle identity $\displaystyle \cos^2{u} = \frac{1+\cos(2u)}{2}$ to help in finding the antiderivative.