CAIE A-Level Mathematics 1.8.4 Area & volume

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

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

How this is tested

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: 5]

Figure for Question 5(b) — CAIE A-Level Mathematics

The equation of a curve is $y=4 x^{\frac{1}{2}}-x$ . The curve has a maximum point when x=a and crosses the x-axis at the point with coordinates (b, 0), where b>0. The shaded region is bounded by the curve, the line x=a and the x-axis (see diagram).

Find the exact area of the shaded region.

How this is tested

Question 5(b)