Evaluate the double integral

Question

Evaluate the double integral ∬R (y^2)/(x^2+y^2) dA, where R is the region that lies between the circles x^2+y^2=25 and x^2+y^2=36, by changing to polar coordinates.

Answer 1

Let move to polar coordinate: x = rcosθ, y = rsinθ, dA = dxdy = rdrdθ.

Now region R of integration: 5 ≤ r ≤ 6 and 0 ≤ θ ≤ 2π.

And because ∫∫Rr²sin²θ/r²·rdrdθ = ∫∫Rsin²θ·rdrdθ = ∫₀^2π(1 - cos2θ)/2dθ·∫₅⁶rdr = 1/2(θ - 1/2sin2θ)₀^2π· r²/2₅⁶ =

1/2·2π·(6² - 5²)/2 = 11π/2