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Past exam of the mathematics course of the University of Oxford

2021

Prelims

Mathematics V

{c}

A <parametrization> of the cylindrical patch is
$$
\mathbf r(x,\theta)=(x,2\cos\theta,2\sin\theta),\qquad
0<\theta<\pi/2,\quad0<x<2\cos\theta.
$$
The last range uses the planes $x=0$ and $x=y$. The <surface area element> is $|\mathbf r_x\times\mathbf r_\theta|\,dx\,d\theta=2\,dx\,d\theta$. Hence
$$
\boxed{A=\int_0^{\pi/2}\int_0^{2\cos\theta}2\,dx\,d\theta=4.}
$$
Including or excluding the boundary curves does not change the <surface area>.

Solution

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

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

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

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