Hello class, I apologize for the unexpected cancellation tonight due to a migraine. To ensure we stay on track for our upcoming sessions, I have compiled a summary of the key concepts from Sections 6-2 and 6-3 based on the attached lecture notes. Please review these materials so we can jump right into problem-solving on Tuesday.

Section 6.2: Volumes by Slicing, Disks, and Washers

In this section, we extend the concept of definite integrals from finding area under a curve to finding the volume of a solid. The fundamental idea is to slice the solid into thin pieces, approximate each slice as a cylinder, and sum them up.

1. The General Slicing Formula
If a solid lies between $x = a$ and $x = b$, and the cross-sectional area at point $x$ is given by $A(x)$, the volume is defined as:

$$V = \int_a^b A(x) dx$$

2. Solids of Revolution (The Disk Method)
When we rotate a region around an axis (like the x-axis), the cross-sections are circles. If the radius of the slice is $R(x)$, the area is $A(x) = \pi[R(x)]^2$.

$$V = \int_a^b \pi [R(x)]^2 dx$$

3. The Washer Method
If the solid has a "hole" in the middle (imagine a donut or a washer), we must subtract the volume of the inner empty space from the outer solid. We identify an outer radius $R_{out}$ and an inner radius $r_{in}$.

$$V = \pi \int_a^b \left( [R_{out}(x)]^2 - [r_{in}(x)]^2 \right) dx$$

Section 6.3: Volumes by Cylindrical Shells

Sometimes, slicing perpendicular to the axis of rotation creates difficult algebra (e.g., trying to solve for $x$ in terms of $y$). In these cases, we use the Method of Cylindrical Shells. Instead of slicing the solid like a loaf of bread, imagine decomposing it into nested cylindrical layers, like the layers of an onion.

The volume of a typical shell is calculated using its circumference, height, and thickness:

  • Circumference: $2\pi r$ (where $r$ is the radius from the axis of rotation)
  • Height: $h$ (the function value, usually $f(x)$)
  • Thickness: $dx$ (or $\Delta x$)

This gives us the formula for rotating around the y-axis:

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

An easy mnemonic to remember this formula is:

$$V = \int [\text{circumference}] \cdot [\text{height}] \cdot [\text{thickness}]$$

Choosing a Method

One of the most important skills in this chapter is deciding which method to use. As detailed in the notes (slide 33), consider the following:

  • Rectangle Orientation: Draw a sample rectangle in your region.
  • If the rectangle is perpendicular to the axis of rotation, you are creating a Disk or Washer (Section 6.2).
  • If the rectangle is parallel to the axis of rotation, you are creating a Cylindrical Shell (Section 6.3).

We will cover these sections in depth on Tuesday, followed by Sections 6-4 and 6-5 on Thursday. Please bring your questions to class!