Question

find the exact area of the regions bounded by y=x^3/(sqrt(4-x^2), y=0,x=0,x=sqrt(2)

Answer 1

\[\int\limits_{0}^{\sqrt{2}} x ^{3}/\sqrt{4-x ^{2}}\]

Answer 2

all we're doing here is integrating up to the line x=sqrt2\[\int_{0}^{\sqrt2}\frac{x^3}{\sqrt{4-x^2}}dx\]which can be done with the substitution\[x=2\sin\theta\to dx=2\cos\theta d\theta\]