 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2/18/b/solution

The region is the triangle

0 \leq y \leq x \leq 1

. Reversing the integration order makes the inner integral elementary:

\int_{0}^{1} \int_{y}^{1} e^{x^{2}} d x d y = \int_{0}^{1} \int_{0}^{x} e^{x^{2}} d y d x = \int_{0}^{1} x e^{x^{2}} d x = \frac{e - 1}{2} .

The integrand is continuous on this compact triangle, so Fubini theorem justifies exchanging the order.

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