Washer Method

Definition

The washer method is a technique used in integral calculus to calculate the volume of a three-dimensional solid by revolving a two-dimensional region around an axis. It involves visualizing the solid as a series of thin circular discs or washers stacked together, with the volume of each washer contributing to the overall volume of the solid.

5 Must Know Facts For Your Next Test

1. The washer method is particularly useful for calculating the volume of solids of revolution that have a circular cross-section, such as cylinders, cones, and spheres.
2. The formula for the volume of a washer is $\pi (r_2^2 - r_1^2) \, dx$, where $r_1$ and $r_2$ are the inner and outer radii of the washer, respectively, and $dx$ is the thickness of the washer.
3. The washer method can be applied in both cylindrical and spherical coordinate systems, depending on the shape of the solid being evaluated.
4. In cylindrical coordinates, the volume of a solid of revolution is calculated by integrating the area of the generating region with respect to the variable representing the distance from the axis of rotation.
5. In spherical coordinates, the volume of a solid of revolution is calculated by integrating the area of the generating region with respect to the variable representing the angle from the positive z-axis.

Review Questions

• Explain how the washer method can be used to calculate the volume of a cone in cylindrical coordinates.
  • To calculate the volume of a cone using the washer method in cylindrical coordinates, we would first visualize the cone as a series of thin circular washers stacked together. The radius of each washer would vary linearly from the base of the cone to the apex, with the innermost washer having a radius of 0 and the outermost washer having a radius equal to the base of the cone. We would then integrate the formula for the volume of a washer, $\pi (r_2^2 - r_1^2) \, dx$, with respect to the variable representing the distance from the axis of rotation, to obtain the total volume of the cone.
• Describe how the washer method can be used to calculate the volume of a sphere in spherical coordinates.
  • To calculate the volume of a sphere using the washer method in spherical coordinates, we would visualize the sphere as a series of thin circular washers stacked together, with each washer representing a cross-section of the sphere at a different angle from the positive z-axis. The radius of each washer would vary based on the distance from the origin and the angle from the positive z-axis. We would then integrate the formula for the volume of a washer, $\pi (r_2^2 - r_1^2) \, d\theta$, with respect to the variable representing the angle from the positive z-axis, to obtain the total volume of the sphere.
• Analyze how the choice of coordinate system (cylindrical or spherical) can affect the application of the washer method when calculating the volume of a solid of revolution.
  • The choice of coordinate system can significantly impact the application of the washer method when calculating the volume of a solid of revolution. In cylindrical coordinates, the washer method is well-suited for solids with circular cross-sections, as the formula for the volume of a washer can be easily integrated with respect to the distance from the axis of rotation. However, in spherical coordinates, the washer method may be more complex, as the formula for the volume of a washer must be integrated with respect to the angle from the positive z-axis. The choice of coordinate system should be based on the shape and symmetry of the solid being evaluated, as this can simplify the integration process and lead to more efficient volume calculations.

Related terms

Disk Method: The disk method is a similar technique to the washer method, but it involves revolving a two-dimensional region around an axis to create a solid of revolution, without the use of washers.

Cylindrical Coordinates:

Cylindrical coordinates are a three-dimensional coordinate system that uses the distance from the origin, the angle around the axis, and the height along the axis to specify a point in space.

Spherical Coordinates:

Spherical coordinates are a three-dimensional coordinate system that uses the distance from the origin, the angle from the positive z-axis, and the angle around the positive x-axis to specify a point in space.