 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v-resit/question-3/a/solution

The annulus is

1 \leq r \leq 2

d x d y = r d r d θ

, so

\iint_{D} e^{- x^{2} - y^{2}} d x d y = \int_{0}^{2 π} \int_{1}^{2} e^{- r^{2}} r d r d θ = π [- e^{- r^{2}}]_{1}^{2} = π (e^{- 1} - e^{- 4}) .

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