 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2007/g/2/i/solution

r (x, θ) = (x, y (x) cos θ, y (x) sin θ), 0 \leq θ \leq 2 π .

The magnitude of the cross product

r_{x} \times r_{θ}

y 1 + y^{'2}

. Integrating the surface area element over

θ

gives

J [y] = 2 π \int_{x_{1}}^{x_{2}} y (x) 1 + y^{'} (x)^{2} d x .

Positivity of

y

removes an otherwise necessary absolute value in the area element.

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