Washer method

Definition

The washer method is a technique used to find the volume of a solid of revolution by integrating the area of circular cross-sections perpendicular to an axis. This method involves slicing the solid into thin disks or washers, where each washer has an inner radius and an outer radius, allowing for the calculation of the volume between two curves. The integration of these areas over a specified interval provides the total volume of the solid formed when a region in a plane is revolved around an axis.

5 Must Know Facts For Your Next Test

1. The washer method is especially useful when dealing with solids that have holes or voids, allowing for accurate volume calculations.
2. When applying the washer method, the volume of each washer is calculated using the formula $$V = \pi(R^2 - r^2)h$$, where R is the outer radius, r is the inner radius, and h is the thickness of the washer.
3. The method requires setting up an integral to sum all the volumes of washers across the specified bounds, typically in terms of x or y depending on the axis of rotation.
4. To implement the washer method correctly, it is crucial to identify the outer and inner functions that define the boundaries of the region being revolved.
5. When integrating, you may need to change limits according to whether you're rotating around the x-axis or y-axis, ensuring proper alignment with the functions involved.

Review Questions

• How does the washer method differ from the disk method when calculating volumes of solids of revolution?
  • The washer method differs from the disk method primarily in that it accounts for solids with hollow sections by using inner and outer radii. While both methods involve integrating cross-sectional areas perpendicular to an axis, the disk method only considers solid disks with no voids. In contrast, the washer method subtracts the area of the inner circle from that of the outer circle, effectively calculating volumes where material is missing.
• Describe how you would set up an integral to use the washer method for a solid formed by rotating a region between two curves around an axis.
  • To set up an integral using the washer method, first identify the two curves that bound the region and determine their intersection points to define your limits of integration. Next, decide whether you are rotating around the x-axis or y-axis, which will dictate whether you express your radii in terms of x or y. The integral can then be set up as $$V = \pi \int_{a}^{b} (R^2 - r^2) \, dx$$ for rotation about the x-axis or $$V = \pi \int_{c}^{d} (R^2 - r^2) \, dy$$ for rotation about the y-axis, where R and r represent the outer and inner functions respectively.
• Evaluate how changing the order of integration affects your calculations when using the washer method for volume.
  • Changing the order of integration can significantly impact your calculations when using the washer method. If you switch from integrating with respect to x to integrating with respect to y, you need to re-evaluate your limits and potentially redefine your outer and inner functions. This might lead to different integrals that capture varying aspects of how regions are defined between curves. Ensuring that you maintain consistency in how you interpret those boundaries is crucial for achieving accurate volume results.

Related terms

disk method:

A technique for calculating volumes of solids of revolution by summing the volumes of solid disks, which are formed by revolving a function around an axis without considering any hollow parts.

cross-section: The intersection of a solid with a plane, which reveals a shape that helps in understanding the structure and calculating its volume.

solids of revolution: Three-dimensional shapes created when a two-dimensional region is rotated around a line (axis), generating various geometric forms based on the shape being revolved.