Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics IV / 3 / b / ii

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- Solution ii

Solution

For the parametrized surface in i,

R_{t} R_{θ} = ((c - a) cos θ, (c - a) sin θ, d - b), = (- ρ sin θ, ρ cos θ, 0) .

These tangent vectors are orthogonal, and their Euclidean norms are

(c - a)^{2} + (d - b)^{2}

and

ρ

, respectively. Their cross product therefore has Euclidean norm

ρ (c - a)^{2} + (d - b)^{2}

. Integration gives

Area (Σ_{1}) = 2 π (c - a)^{2} + (d - b)^{2} \int_{0}^{1} (a + (c - a) t) d t = π (a + c) (a - c)^{2} + (b - d)^{2} .

This is the lateral surface area of a conical frustum, with no end disks included.

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