 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/6/solution

The integration region is the triangle

0 \leq y \leq x \leq π

. Reversing its order gives

I = \int_{0}^{π} \int_{0}^{x} \frac{s i n x}{x} d y d x = \int_{0}^{π} sin x d x = 2.

The factor of

x

from the inner interval cancels the denominator. At

x = 0

sin x / x

has a removable singularity, so no divergent integral is being hidden by this change of order. Fubini's theorem applies to its continuous extension on the bounded triangle.

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