A Disk

Lingual

Mathematics

Calculus

Disk Method integration

The Disk Method (Perpendicular Slices, Solid Core) Used when rotating a single curve flush against the axis of rotation. The cross-section is a solid circular disk with radius \(R\) and thickness \(dx\) or \(dy\).

• Rotation about a horizontal axis (\(x\)-axis or \(y = k\)):

\[V = \pi \int_{a}^{b} [R(x)]^2 \, dx\]

Where \(R(x) = f(x) - k\) (or simply \(f(x)\) if rotating around \(y=0\)).

• Rotation about a vertical axis (\(y\)-axis or \(x = h\)):

\[V = \pi \int_{c}^{d} [R(y)]^2 \, dy\]