AHL 5.12 (HL)—Areas and volumes

Rotating a region about an axis creates thin discs, washers or shells. About the x-axis, a disc method uses V=π∫[radius]²dx; washers subtract the inner radius squared from the outer.

Choose the variable and axis so the radius and bounds are explicit. If the region crosses the axis or the radii change order, split the integral rather than silently using one formula.

Rotating y=x on 0≤x≤2 about the x-axis gives V=π∫₀²x²dx=8π/3. The radius is y=x, not the horizontal coordinate itself by definition.

An area integral is not automatically a volume. Square the radius, include π, and check whether the axis and region create overlapping solids.

For area between a curve and an axis, split at intersections and use absolute geometric pieces when the question asks for area because an integral below the axis is negative. For revolution about the y-axis, $V=\pi\int_a^b x^2\,dy$; about the x-axis, $V=\pi\int_a^b y^2\,dx$. Express both bounds and radius in the integration variable.