AHL 5.17 (HL)—Areas and volumes of revolution

Volumes of revolution depend on slices and axis choice.

Rotating a region creates disks, washers or shells; the radius and thickness must match the axis and variable of integration.

Example

Rotating y=x from 0 to 1 about the x-axis gives V=π∫₀¹x²dx=π/3.

Sketch the region, identify outer/inner radius or shell height, and state the limits.

Using a radius measured from the wrong axis can produce a plausible but incorrect volume.

Area between $x=f(y)$ and the $y$-axis from $y=c$ to $y=d$ is $A=\int_c^d|f(y)|\,dy$, splitting where the curve crosses the axis. Rotation about the $x$-axis gives $V=\pi\int_a^b(R(x)^2-r(x)^2)\,dx$; about the $y$-axis use radii expressed in $y$, $V=\pi\int_c^d(R(y)^2-r(y)^2)\,dy$. Sketch the region so the radius is measured perpendicular to the chosen axis.