12.3#8

The region goes from pi/2 to 3pi/2 with radius' of 3 and 5 so wouldn't
those be my limits of integration? I am unsure of what to do since this
didnt work.

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There's nothing wrong with your limits of integration. The thing you did wrong here is putting the (rcos(t), rsin(t)) into the dA, instead of leaving it with the f where it belongs.