 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/12/a/iv/solution

For a vertical slice through the triangle,

0 \leq x \leq 2

and

0 \leq y \leq x /2

. Reversing the order by Fubini's theorem gives

\int_{0}^{2} (\int_{0}^{x /2} x^{2} y^{2} d y) d x = \int_{0}^{2} \frac{x ^{5}}{24} d x = \frac{4}{9} .

The agreement checks that both orders describe the same region.

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