Frequently Asked Questions: Volume by Disks and Washers
The Disk and Washer methods are techniques in calculus used to find the volume of a solid of revolution. These solids are formed by revolving a 2D region around an axis. The methods involve slicing the solid into thin cross-sections perpendicular to the axis of revolution and summing up the volumes of these slices using integration.
The Disk Method is a specific case of volume by slicing used when the region being revolved is directly adjacent to the axis of revolution. When sliced perpendicular to the axis, each slice is a thin disk (a cylinder). The volume of each disk is π × (radius)² × (thickness).
To use it, you identify the radius R of the disk as the distance from the axis of revolution to the outer boundary of the region. The integral becomes the sum of these disk volumes:
Volume = π ∫ab [R(x)]² ⅆx (if revolving around x-axis or parallel)
Volume = π ∫cd [R(y)]² ⅆy (if revolving around y-axis or parallel)
The Washer Method is used when there is a gap between the region being revolved and the axis of revolution. When sliced perpendicular to the axis, each slice is a thin washer (a disk with a hole in the center). The volume of each washer is the volume of the outer disk minus the volume of the inner disk, times the thickness: π × [(Outer Radius)² − (Inner Radius)²] × (thickness).
To use it, you identify the Outer Radius (R) and Inner Radius (r) as the distances from the axis of revolution to the outer and inner boundaries of the region, respectively. The integral is:
Volume = π ∫ab [[R(x)]² − [r(x)]²] ⅆx (revolving around x-axis or parallel)
Volume = π ∫cd [[R(y)]² − [r(y)]²] ⅆy (revolving around y-axis or parallel)
You use the **Disk Method** when the region being revolved is flush against the axis of revolution, creating a solid shape with no hole in the center when sliced.
You use the **Washer Method** when there is a space between the region being revolved and the axis of revolution, resulting in a solid with a hole through its center (like a washer) when sliced. This gap is often created when revolving the area between two curves around an axis that doesn't pass through the region.
You use the **Washer Method** when there is a space between the region being revolved and the axis of revolution, resulting in a solid with a hole through its center (like a washer) when sliced. This gap is often created when revolving the area between two curves around an axis that doesn't pass through the region.
The choice depends on the axis of revolution and how the functions are defined:
- If revolving around the **x-axis** or a horizontal line, you generally integrate with respect to **x (dx)**. The slices are vertical disks or washers, and their thickness is Δx. You need functions in terms of x, i.e., y = f(x).
- If revolving around the **y-axis** or a vertical line, you generally integrate with respect to **y (dy)**. The slices are horizontal disks or washers, and their thickness is Δy. You need functions in terms of y, i.e., x = f(y).
Yes, the concept of volume by slicing is more general than just disks and washers. If you can find a formula A(x) or A(y) for the area of a cross-section of a solid perpendicular to the x-axis or y-axis, respectively, then the volume of the solid can be found by integrating that area function:
Volume = ∫ab A(x) ⅆx or Volume = ∫cd A(y) ⅆy
Disks and washers are simply specific shapes of these cross-sections (circles or rings).
