How will you prove the following?

Question

#int int_R# #(x^2y^2)/(x^2 +y^2) dxdy = (65pi)/16#,
where, R is the annulus between #x^2+y^2=4# and #x^2+y^2=9#

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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 #