A horizontal strip adjoining the vertical axis sweeps out a disk. Its radius is a horizontal distance, so express the radius in terms of \(y\).

\(V=\pi\int_c^d[\mathrm{g}(y)]^2\,\mathrm{d}y\), where \(x=\mathrm{g}(y)\).

For the region between \(x=y^2\), the vertical axis and \(y=2\), the radius is \(y^2\), not \(y\).

Matching radius and variable

\(V=\pi\int_0^2(y^2)^2\,\mathrm{d}y=\pi\left[\dfrac{y^5}{5}\right]_0^2=\dfrac{32\pi}{5}\).

When a square root appears in the curve equation, rearrange directly for \(x^2\) if possible. This often avoids unnecessary radicals.

On a calculator paper, keep an exact multiple of \(\pi\) when the question requests an exact answer.

Volume of Revolution about the y-axis

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Axis of rotation	Strip	Radius squared	Limits
Horizontal	Vertical	\(y^2\), expressed in \(x\)	\(x\)-coordinates
Vertical	Horizontal	\(x^2\), expressed in \(y\)	\(y\)-coordinates