Evaluate the following iterated integral by converting to polar coordinate system: ∫∫(sin(x^2+y^2) dydx. The bounds are: y=(√9-x^2), y=0, x=3,x=-3

Answer 1

In polar coordinates x = rcosφ, y = rsinφ, x² + y2 = r². dA = dxdy = rdrdφ

Region is: 0≤r≤3, 0≤φ≤π.Our integral I = ∫∫sinr²rdrdφ =π∫sinr²rdr = π/2(-cosr²) with substitution from 0 to 3

I = -π/2(cos9 - cos0) = π/2(1-cos9)