 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/2/solution

x = (v - u) /2

y = (v + u) /2

1/2

, and the transformed triangle is

0 \leq v \leq 3

- v \leq u \leq v

. Hence

I = \frac{1}{2} \int_{0}^{3} \int_{- v}^{v} e^{u / v} d u d v = \frac{1}{2} \int_{0}^{3} v (e - e^{- 1}) d v = \frac{9}{4} (e - e^{- 1}) = \frac{9}{2} sinh 1.

At the single vertex

v = 0

, the printed integrand is undefined. This causes no integral divergence: throughout the remaining triangle

- 1 \leq u / v \leq 1

, so the integrand is bounded between

e^{- 1}

and

e

. Giving the vertex any finite value, or taking a limiting integral with

v > 0

, gives the same answer.

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