Pearson Edexcel IAL Mathematics P4.6.1 Evaluation of volume of revolution

Practise setting up and evaluating volumes of revolution from curves or parametric equations, including exact answers and model contexts.

Exam points

use π∫y^2 dx with correct limits when a shaded region is rotated about the x-axis
convert parametric curves using dx/dt before integrating y^2 dx for volume
evaluate exact volume integrals involving roots, logs, trig functions or exponentials

[Maximum number: 10]

Figure 3

Figure 3 shows a sketch of the curve C with parametric equations

x=6 t-3 \sin 2 t \quad y=2 \cos t \quad 0 \leqslant t \leqslant \frac{\pi}{2}

The curve meets the y-axis at 2 and the x-axis at k, where k is a constant.

(a)

Show that the volume of this solid is given by

\int_{0}^{\alpha} \beta(1-\cos 4 t) \mathrm{d} t

where $\alpha$ and $\beta$ are constants to be found.

[ 4 ]

Attempts $y^{2} \frac{\mathrm{~d} x}{\mathrm{~d} t}=4 \cos ^{2} t(6-6 \cos 2 t)$

M1 A1

$=(2 \cos 2 t+2)(6-6 \cos 2 t)=6\left(2-2 \cos ^{2} 2 t\right)=6(2-(1+\cos 4 t))$

dM1

Volume $=\int_{0}^{\frac{\pi}{2}} \pi y^{2} \frac{\mathrm{~d} x}{\mathrm{~d} t} \mathrm{~d} t=\int_{0}^{\frac{\pi}{2}} 6 \pi(1-\cos 4 t) \mathrm{d} t$

A1

(b)

Hence, using algebraic integration, find the exact volume of this solid.

[ 6 ]

$\left[6 \pi t-\frac{6 \pi \sin 4 t}{4}\right]_{0}^{\frac{\pi}{2}},=3 \pi^{2}$

M1, A1

(6)

(14 marks)

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P4.6.1 - Evaluation of volume of revolution question 1

Question (a)

Question (b)