 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/19/b/solution

The sloping height relation on the cone is

z = b (1 - r / a)

. A parametric equation of its curved surface is therefore

R (r, θ) = (r cos θ, r sin θ, b (1 - r / a)), 0 \leq r \leq a, 0 \leq θ < 2 π .

R_{r} = (cos θ, sin θ, - b / a)

and

R_{θ} = r (- sin θ, cos θ, 0)

d r

and

d θ

produces the two infinitesimal sides requested in the paper. Their cross product is the vector area element:

d S = (R_{r} \times R_{θ}) d r d θ = r (\frac{b}{a} cos θ, \frac{b}{a} sin θ, 1) d r d θ .

Its radial component is positive, so this normal vector points outwards from the solid cone. Taking its norm gives the scalar area element

d S = r 1 + \frac{b ^{2}}{a ^{2}} d r d θ .

The parametric equation degenerates at the apex, but that single point contributes no area.

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