CAIE A-Level Mathematics AS 1.8.4 Area and Volume Questions

Practise using integration to find shaded areas and volumes of revolution from curves, lines, axes, intersections and tangents.

Exam points

use vector notation and operations
solve geometric problems with vectors
interpret vector relationships

[Maximum number: 3]

Figure for Question CAIE A-Level Mathematics AS 1.8.4 Area and Volume Questions question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\sqrt{2 x^{3}+10}$ .

The region shaded in the diagram is enclosed by the curve and the straight lines x=1, x=3 and y=0.

Find the volume of the solid obtained when the shaded region is rotated through $360^{\circ}$ about the x-axis.

State or imply volume is $\pi \int\left(2 x^{3}+10\right) \mathrm{d} x$
Implied if $\pi$ appears only at the end.

Marking guidance:

Do not allow an unsimplified:
$\pi \int\left(\left(2 x^{3}+10\right)^{1 / 2}\right)^{2}$

Integrate to obtain $k_{1} x^{4}+k_{2} x$ and evaluate using limits 1 and 3
Where $k_{1} k_{2} \neq 0$ .

$60 \pi$
OE

Allow from a correct integral and sight of limits. Allow numerical answers in the range 188-189.

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CAIE A-Level Mathematics AS 1.8.4 Area and Volume Questions question 1