Question

Evaluate the integral ${\int }_0^{4a}{\int }_{\frac{y^2}{4a}}^y\left(\frac{x^2-y^2}{x^2+y^2}\right)dxdy$ by changing into polar coordinates.

Answer 1

For protons co-ordinates

$x=a{t}^2$

$y=2at$

$\underset{0}{\overset{2}{\int }}\underset{t=y/2a}{\overset{2}{\int }}\frac{a^2+4-4a^2{t}^2}{a^2{t}^4+4a^2{t}^2}2adt.$

$\underset{0}{\overset{2}{\int }}\underset{y/2a}{\overset{2}{\int }}\frac{t^2-4}{t^2+4}4a^{2}tdndx$

$Whichgivesanswer=\frac{8a^2}{3}$