Welcome back to class! In tonight's session, we are moving beyond simple areas and entering the 3rd dimension. We will be covering Section 6-2 and Section 6-3 (corresponding to Chapter 5 in our text), focusing on how to use definite integrals to calculate the volumes of complex solids.

Below you will find the links to the Zoom meeting and all the lecture materials you need for this week.

Zoom Meeting Link | Class Notes

Section 6-2: Volumes by Slicing, Disks, and Washers

In this section, we define the volume of a solid $S$ by slicing it into thin cross-sections. If we know the area of a cross-section $A(x)$ perpendicular to the x-axis, the volume is defined as:

$$V = \lim_{n \to \infty} \sum_{i=1}^{n} A(x_i^*) \Delta x = \int_a^b A(x) dx$$

Most of the solids we deal with are Solids of Revolution, obtained by rotating a region around an axis. We have two main approaches here:

  • The Disk Method: Used when the solid is solid all the way through (no holes). The cross-sections are circles. $$A = \pi (\text{radius})^2$$
  • The Washer Method: Used when the solid has a hole in the middle. The cross-sections look like washers (rings). $$A = \pi (\text{outer radius})^2 - \pi (\text{inner radius})^2$$

Class Note Highlight: Check the attached handwritten notes for Example 4, where we rotate the region between $y=x$ and $y=x^2$ around the x-axis using the washer method!

Section 6-3: Volumes by Cylindrical Shells

Sometimes, slicing perpendicular to the axis of rotation makes the algebra incredibly difficult (or impossible). In these cases, we use the Method of Cylindrical Shells. Instead of slicing the solid like a loaf of bread, imagine peeling it like an onion.

When rotating around the y-axis, we use vertical cylindrical shells defined by the formula:

$$V = \int_a^b 2\pi x f(x) dx$$

A helpful mnemonic to remember the integrand for shells is:

Volume = $\int$ [Circumference] $\times$ [Height] $\times$ [Thickness]

Disks/Washers vs. Shells: How to Choose?

One of the biggest challenges in this chapter is deciding which method to use. Here is a quick guide based on our lecture slides:

  1. Draw a sample rectangle (slice) in your region.
  2. Determine orientation:
    • If your slice is perpendicular to the axis of rotation, use Disks/Washers.
    • If your slice is parallel to the axis of rotation, use Cylindrical Shells.

Be sure to review the handwritten class notes attached above, specifically Example 1 (Shells) and the comparison in Example 3 where we verify results. Good luck studying, and see you in the Zoom meeting!

Files:
Section 6-2: Powerpoint | PDF
Section 6-3: Powerpoint | PDF