The disk method is a calculus technique for finding the volume of a solid formed by rotating a flat region around an axis. It matters because many rounded objects, such as bowls, lenses, machine parts, and containers, can be modeled as solids of revolution. Instead of trying to measure the whole 3D shape at once, the method slices it into many thin circular disks.
Adding the volumes of all those disks with an integral gives the total volume.
Key Facts
- Disk method around the x-axis: V = π ∫ from a to b [f(x)]^2 dx
- Each thin disk has radius r = f(x) and thickness dx
- Volume of one thin disk is dV = πr^2 dx
- Disk method around the y-axis: V = π ∫ from c to d [g(y)]^2 dy
- Use disks when the rotated region touches the axis of rotation, so there is no hole
- Units of volume are cubic units, such as cm^3, m^3, or ft^3
Vocabulary
- Solid of revolution
- A 3D shape made by rotating a 2D region around a line called an axis of rotation.
- Disk method
- A volume method that adds the volumes of many thin circular slices perpendicular to the axis of rotation.
- Cross-section
- A flat slice of a solid taken perpendicular or parallel to a chosen axis.
- Radius function
- The function that gives the radius of each disk at a given position along the axis.
- Definite integral
- An integral over a closed interval that represents the accumulated total of a changing quantity.
Common Mistakes to Avoid
- Forgetting to square the radius is wrong because disk area is A = πr^2, not A = πr.
- Using the wrong variable of integration is wrong because rotating around the x-axis usually uses dx, while rotating around the y-axis usually uses dy.
- Including π twice is wrong because the formula V = π ∫ [radius]^2 already contains the circle area constant.
- Choosing the diameter instead of the radius is wrong because the disk method requires the distance from the axis of rotation to the curve, not the full width across the solid.
Practice Questions
- 1 Find the volume formed by rotating the region under y = 2x from x = 0 to x = 3 around the x-axis.
- 2 Find the volume formed by rotating the region under y = sqrt(x) from x = 0 to x = 4 around the x-axis.
- 3 A region between a curve and the x-axis is rotated around the x-axis. Explain why the disk method uses circular cross-sectional area rather than the original 2D area under the curve.