Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics IV / 2 / c

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- Solution c

Solution

Let

D

be the filled projected ellipse. The part of the plane bounded by

C

is the parametrized surface

R (x, y) = (x, y, 1 - x - y), (x, y) \in D .

Its tangent vectors satisfy

R_{x} = (1, 0, - 1), R_{y} = (0, 1, - 1), ∥ R_{x} \times R_{y} ∥ = ∥ (1, 1, 1) ∥ = 3 .

The surface area is therefore

A (C) = \iint_{D} 3 d x d y = 3 \frac{8 π}{3 3} = \frac{8 π}{3} .

This also illustrates the orthogonal projection area formula: the unit normal to the plane has vertical component

1/ 3

, so projection multiplies area by

1/ 3

.

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