How will you prove the following?

Question

$\int \int_{R}$ $int int_R$ $\frac{x^{2} y^{2}}{x^{2} + y^{2}} d x d y = \frac{65 π}{16}$ $(x^2y^2)/(x^2 +y^2) dxdy = (65pi)/16$ ,
where, R is the annulus between $x^{2} + y^{2} = 4$ $x^2+y^2=4$ and $x^{2} + y^{2} = 9$ $x^2+y^2=9$

952 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-17T08:36:20

See below

Write it in polar.

$int int_R \ (x^2y^2)/(x^2 +y^2) \ color(blue)(dx\ dy)$

$equiv int_0^(2 pi) int_2^3 (r^2 cos^2 theta \ r^2 sin^2 theta )/(r^2) color(red)(\ r \ dr \ d theta)$

$=1/4 int_0^(2 pi) int_2^3 \ r^3 sin^2 (2 theta) \ dr \ d theta$

$=1/4 int_0^(2 pi) ( \ r^4/4)_2^3 sin^2 (2 theta) \ d theta$

$=65/16 int_0^(2 pi) (1 - cos(4 theta))/2 \ d theta$

$=65/32 ( theta - 1/4 sin(4 theta) )_0^(2 pi)$

$=65/16 pi$