Sajjad Bin Samad
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- Joined
- Mar 8, 2020
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Evaluate the following:-
[MATH]\int _0^4\int _{-\sqrt{4x-x^2}}^{\sqrt{4x-x^2}}\:\:\left(x^2+y^2\right)dy\:dx[/MATH]
Since the function is even,
[MATH]2\int _0^4\left[\int _0^{\sqrt{4x-x^2}}\:\:\left(x^2+y^2\right)dy\:\right]dx[/MATH]
After this integrating with respect to y and evaluating at y=0 to y= [MATH]\sqrt{4x-x^2}[/MATH]
we get,
[MATH]\frac{2}{3}\int _0^4\left(\sqrt{4x-x^2}\right)\left(3x^2+4x-x^2\right)\:dx[/MATH]
Beyond this, I can not follow up with my textbook. But what I see in the book is that first, they convert the coordinates to polar ( don't know why or how ) then they use beta and gamma function and got the answer which is 25pi.
Step by step solution is appreciated and also suggest me a book which is very thorough.
[MATH]\int _0^4\int _{-\sqrt{4x-x^2}}^{\sqrt{4x-x^2}}\:\:\left(x^2+y^2\right)dy\:dx[/MATH]
Since the function is even,
[MATH]2\int _0^4\left[\int _0^{\sqrt{4x-x^2}}\:\:\left(x^2+y^2\right)dy\:\right]dx[/MATH]
After this integrating with respect to y and evaluating at y=0 to y= [MATH]\sqrt{4x-x^2}[/MATH]
we get,
[MATH]\frac{2}{3}\int _0^4\left(\sqrt{4x-x^2}\right)\left(3x^2+4x-x^2\right)\:dx[/MATH]
Beyond this, I can not follow up with my textbook. But what I see in the book is that first, they convert the coordinates to polar ( don't know why or how ) then they use beta and gamma function and got the answer which is 25pi.
Step by step solution is appreciated and also suggest me a book which is very thorough.