 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/15/b/i/solution

Here

y \geq 0

, so the absolute value equals

y

J = \int_{0}^{1} \int_{0}^{x} \int_{0}^{x y} x y d z d y d x .

The integrand is constant with respect to

z

J = \int_{0}^{1} \int_{0}^{x} x^{2} y^{2} d y d x = \frac{1}{3} \int_{0}^{1} x^{5} d x = \frac{1}{18} .

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