Evaluate the I?

Question

This is question is inspired from this

find the value of $$I=\int_{0}^{2\pi} \int_{0}^{\sqrt 2 a}\sqrt{ \frac{u^4}{a^2} +u^2} du dv$$

My attempt : $$I=\int_{0}^{2\pi} \int_{0}^{\sqrt 2 a}u\sqrt{ \frac{u^2}{a^2} +1} du dv$$

after that im not able to solved this

Answer 1

Before I worked out a complete solution, @TitoEliatron has left a key step in his comment.

\begin{align} I&=\int_{0}^{2\pi} \int_{0}^{\sqrt 2 a}u\sqrt{ \frac{u^2}{a^2} +1} \,du \, dv\\ &=\int_0^{2\pi}dv \; \frac12 \int_0^{\sqrt2 a} \sqrt{ \frac{u^2}{a^2} +1} \, d(u^2) \\ &= \pi \int_0^{2a^2} \sqrt{x/a^2 + 1} \, dx \tag{$x = u^2$} \\ &= \frac{\pi}{a} \int_0^{2a^2} \sqrt{x + a^2} \, dx \\ &= \frac{2\pi}{3a} \left[ (x+a^2)^{3/2} \right]_0^{2a^2} \\ &= \frac{2\pi (3\sqrt3 - 1) a^2}{3} \end{align}

Edit : The $a^3$ in the numerator cancelled with $a$ in the denominator to give $a^2$.

Answer 2

For

$$\int_{0}^{\sqrt 2 a}u\sqrt{ \frac{u^2}{a^2} +1} du $$

use the substitution $t= \frac{u^2}{a^2}+1$. Then you get the integral

$$\frac{a^2}{2}\int_1^3 \sqrt{t} dt,$$

which is independent of $v$, hence

$$\int_{0}^{\sqrt 2 a}u\sqrt{ \frac{u^2}{a^2} +1} du = 2 \pi \cdot \frac{a^2}{2}\int_1^3 \sqrt{t} dt.$$