Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 1 / c

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- Solution c

Solution

A parametrization of the cylindrical patch is

r (x, θ) = (x, 2 cos θ, 2 sin θ), 0 < θ < π /2, 0 < x < 2 cos θ .

The last range uses the planes

x = 0

and

x = y

. The surface area element is

∣ r_{x} \times r_{θ} ∣ d x d θ = 2 d x d θ

. Hence

A = \int_{0}^{π /2} \int_{0}^{2 c o s θ} 2 d x d θ = 4.

Including or excluding the boundary curves does not change the surface area.

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