 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iv/3/b/ii/solution

R (x, ϕ) = (x, f (x) cos ϕ, f (x) sin ϕ), a < x < b, 0 \leq ϕ < 2 π .

Its tangent vectors are orthogonal, with norms

1 + f^{'} (x)^{2}

and

∣ f (x) ∣

. Thus

A = 2 π \int_{a}^{b} ∣ f (x) ∣ 1 + f^{'} (x)^{2} d x .

The absolute value is necessary if the function graph passes below the axis. The formula gives the curved surface area; it adds no end discs. Zeros of

f

can be treated by splitting the integral into regular portions and taking limits.

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