Integration on y=x(2ax -x^2)^1/2 wrt to x where a is a constant

[needMAthhelps]

currently having problem integrating above function shown. Do anyone know the steps to tackle this kind of integration problem?

 

[MarkFL]

Hello, and welcome to FMH!

I would try completing the square on the radicand:

[MATH]I=\int_0^a x\sqrt{a^2-(x-a)^2}\,dx[/MATH]
Now, let:

[MATH]x-a=au\implies dx=a\,du[/MATH]
And we have:

[MATH]I=a\int_{-1}^0 (u+1)\sqrt{1-u^2}\,du[/MATH]
Can you proceed?

 

[needMAthhelps]

OH did not see that! thanks for the help!! Super helpful

 

[MarkFL]

To follow up:

[MATH]I=\frac{3\pi-4}{12}a=888\implies a=\frac{10656}{3\pi-4}\approx1964.32[/MATH]