Edexcel A-Level Mathematics A2 P4.6.1 Evaluation of Volume of Revolution Questions

Practise Edexcel IAL P4.6.1 volumes of revolution: set correct limits, convert parametric curves, and evaluate exact π∫y² dx integrals.

Exam points

Set up π∫y^2 dx with correct limits for a region rotated about the x-axis
Convert parametric curves using dx/dt before integrating y^2 dx
Evaluate the volume integral exactly, including roots, logs, trigonometric or exponential terms

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

(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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Edexcel A-Level Mathematics A2 P4.6.1 Evaluation of Volume of Revolution Questions question 1

Question (a)

Question (b)