Key takeaway: The disk method finds the volume of a solid formed by rotating a region about an axis by adding up thin circular disks, each with area pi times radius squared.
Grade 12 – AP Calculus ABAP Calculus AB Unit 8Calculus
Definition of Disk Method
The disk method computes the volume of a solid of revolution. Rotate a region around a line that forms one of its edges, and slice the resulting solid perpendicular to the axis. Every slice is a thin circular disk, a cylinder with tiny thickness.
A disk’s volume is the area of its circular face times its thickness. Adding all disks along the axis is a definite integral. The radius of each disk is the distance from the axis to the curve, so it changes with position.
Formula and notation
Rotating \(y=R(x)\) about the \(x\)-axis between \(x=m\) and \(x=n\):
\[ V=\pi\int_{m}^{n}\big[R(x)\big]^2\,dx. \]
The radius \(R(x)\) is squared inside the integral, and \(\pi\) comes outside. When the region does not touch the axis, the washer method subtracts a hole.
Key facts
- The radius is the distance from the axis of rotation to the curve, not the curve’s height in general.
- In the disk method, square the radius before integrating, never after.
- Use dy and functions of y when rotating around a vertical line.
- Volume units are cubic units; the integral of an area gives a volume.
Disk Method example
The region under \(y=\sqrt{3x}\) from \(x=0\) to \(x=6\) is rotated about the \(x\)-axis. Find the volume of the solid.
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The radius of each disk is \(R(x)=\sqrt{3x}\), so \([R(x)]^2=3x\).
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Set up: \(V=\pi\int_{0}^{6}3x\,dx\).
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Integrate: \(\pi[\frac{3x^2}{2}]_{0}^{6}=\pi\cdot\frac{3\cdot 36}{2}=54\pi\).
The volume is \(54\pi\approx 169.6\) cubic units.
Common mistake
Watch out
A student forgets to square the radius and evaluates \(\pi\int_{0}^{6}\sqrt{3x}\,dx=12\sqrt{2}\,\pi\approx 53.3\).
Why it is wrong
Each disk has an area \(\pi r^2\), so the radius must be squared. Without the square, the integrand has the wrong units for an area and the volume is far too small. With \(r^2=3x\), the disk method gives \(54\pi\approx 169.6\).
Related terms and lessons
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