Disk method

Definition

The disk method is a technique used to find the volume of a solid of revolution by integrating the area of circular disks. It involves slicing the solid perpendicular to the axis of rotation, where each slice represents a thin disk, and then summing the volumes of these disks using integration. This method is particularly useful when the solid is generated by rotating a function around a horizontal or vertical axis.

5 Must Know Facts For Your Next Test

1. The volume of each disk is calculated using the formula $$V = \pi r^2 h$$, where $$r$$ is the radius and $$h$$ is the thickness of the disk.
2. The disk method requires the function being rotated to be expressed in terms of $$y$$ when rotating around the x-axis, or in terms of $$x$$ when rotating around the y-axis.
3. To find the total volume, you integrate the area of all disks over the interval defined by the bounds of rotation.
4. The limits of integration correspond to the points where the solid starts and ends along the axis of rotation.
5. The disk method can be visualized as stacking many thin circular disks along the axis, which collectively form the solid.

Review Questions

• How does the disk method differ from other volume calculation methods like the washer method?
  • The disk method is specifically used for finding volumes of solids that are completely filled with material, utilizing circular cross-sections. In contrast, the washer method is applied to solids with holes or gaps, requiring an inner and outer radius for each cross-section. While both methods involve integration, they differ in how they handle the geometry of the solid being analyzed.
• Demonstrate how to set up an integral using the disk method for a function rotated about the x-axis.
  • To set up an integral using the disk method for a function $$f(x)$$ rotated about the x-axis, identify the limits of integration $$a$$ and $$b$$ where you want to compute the volume. The volume integral can be expressed as $$V = \int_{a}^{b} \pi [f(x)]^2 \, dx$$. This integral sums up the volumes of all infinitesimally thin disks from $$x = a$$ to $$x = b$$, giving you the total volume of the solid formed by rotation.
• Evaluate how changing the axis of rotation impacts the application of the disk method.
  • Changing the axis of rotation significantly alters how we set up our integrals in the disk method. For instance, if we switch from rotating around the x-axis to rotating around the y-axis, we must express our function in terms of $$y$$ instead. This switch affects both the limits of integration and how we calculate radii for our disks. It’s essential to re-evaluate which variables are dependent on others when adjusting axes to ensure accurate volume calculations.

Related terms

washer method: A variation of the disk method that accounts for hollow solids, where each cross-section consists of an outer disk and an inner hole.

solid of revolution: A three-dimensional shape obtained by rotating a two-dimensional shape around an axis.

definite integral: An integral that calculates the accumulation of quantities, such as area or volume, over a specified interval.