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Mathematics course of the University of Oxford

Past exam of the mathematics course of the University of Oxford

2015

Prelims

Mathematics V

{c}

At a fixed $z\in[-1,1]$, both <cylinder> inequalities become
$$
|x|\leq\sqrt{1-z^2},\qquad |y|\leq\sqrt{1-z^2}.
$$
Thus the <cross section> is a <square> of side $2\sqrt{1-z^2}$ and <area> $4(1-z^2)$. Integrating these <cross-sectional areas> gives the <volume> of this <bicylinder>:
$$
\boxed{\operatorname{Vol}(R_1)=\int_{-1}^{1}4(1-z^2)\,dz
=4\left[z-\frac{z^3}{3}\right]_{-1}^{1}=\frac{16}{3}.}
$$

Solution

Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 3 / b / Solution

Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 3 / b / Solution

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Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 3 / b / Solution

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