Evaluate ∫∫ A sin(x^2 + y^2) dA, where A is the region bounded between x^2 + y^2 = 4 and x^2 + y^2 = 9. please use polar coordinates

Answer 1

In the transformation from cartesian to polar coordinates r² = x² + y²

The bounds of integration are particularly nice because they don't depend on θ , the angle. Therefore, the integration is from r² = 4 to 9. This has the solutions for r positive or negative, but if we allow the angle to go from 0 to 2pi, we can state the limits as r = [2,3] and θ = [0,2pi]

The integral becomes (I'm assuming the A is under the integral sign indicating the limits)

Int from 0 to 2π(Int from 2 to 3(sin(r²) rdr)dθ)

Note that rdrdθ is the areal element in polar coordinates.

Nothing depends on θ, so we can do that integral first

Integral = 2π (Int from 2 to 3 of (sin(r²) rdr) ) which is integrable as 1/2sin(u) du

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