Question

Solve $\frac{{\int }_0^a{x}^4\sqrt{a^2-x^2}}{{\int }_0^a{x}^2\sqrt{a^2-x^2}}$

Answer 1

Let $I=\frac{\underset{0}{\overset{a}{\int }}{x}^4\sqrt{a^2-x^2}dx}{\underset{0}{\overset{a}{\int }}{x}^2\sqrt{a^2-x^2}dx}$

$=\underset{0}{\overset{a}{\int }}\frac{x^4\sqrt{a^2-x^2}}{x^2\sqrt{a^2-x^2}}dx$

$=\underset{0}{\overset{a}{\int }}{x}^{2}dx$

$={\left[\frac{x^3}{3}\right]}_0^a+c$

$=\frac{a^3}{3}+c$