if D =x^2+y^2<a^2 then :

Question

∫∫ ΙxΙ dA

Answer 1

Via symmetry, we can double the result from when cosθ >= 0, or -π/2 < θ < π/2, 0 <= r <= a

Use polar coordinates, and recall that r dr dθ = dy dx and s = r cosθ

We use I[a,b] for the integral from a to b and E[a,b] for the evaluation from a to b

2 I[0,a] I[-π/2,π/2] r^2 cos θ dθ dr =

2 I[0,a] r^2 sin θ E[-π/2,π/2] dr =

4 I[0,a] r^2 dr =

4/3r^3 E[0,a] =

4/3 a^3