CAIE A-Level Mathematics 1.8.4 Area & volume

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

Exam points

set correct bounds from intersections, axes or tangents before integrating a region
subtract curve-line or curve-curve integrals with the correct order for shaded area
for revolution volumes, choose x or y integration to match the axis and squared radius

[Maximum number: 4]

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

The diagram shows the curve with equation $y=9\left(x^{-\frac{1}{2}}-4 x^{-\frac{3}{2}}\right)$ . The curve crosses the x-axis at the point A.

Find the area of the region bounded by the curve, the x-axis and the line x=9.

$\int 9\left(x^{-\frac{1}{2}}-4x^{-\frac{3}{2}}\right)\,dx =9\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}-\frac{4x^{-\frac{1}{2}}}{-\frac{1}{2}}\right)$
B2,1,0: B2 for all 3 terms correct: 9, $\frac{x^{1/2}}{1/2}$ , and $\frac{-4x^{-1/2}}{-1/2}$ ; B1 for 2 of the 3 terms correct.
$9\left[\left(6+\frac{8}{3}\right)-(4+4)\right]$
M1 Apply limits $4\rightarrow 9$ to an integrated expression with no consideration of other areas.
A1 6. Use of $\pi$ scores A0.

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1.8.4—Area and volume question 1