Welcome back to Professor Baker's Math Class! In this lesson, we are taking our integration skills into the third dimension. We aren't just finding the area under a curve anymore; we are taking that area, rotating it around an axis, and calculating the Volume of the resulting solid.

In Sections 6-2 and 6-3, we cover two primary techniques for finding these volumes: the Disk/Washer Method and the Cylindrical Shell Method.

Section 6-2: The Disk and Washer Methods

The definition of volume starts with the idea of slicing a solid into infinitely thin cross-sections. If we know the area of a cross-section, $A(x)$, we can integrate it over an interval $[a, b]$ to find the volume:

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

When we rotate a region around an axis, the cross-sections are circles (disks) or rings (washers).

  • Disk Method: Use this when the solid is solid all the way through (no gaps). The area is simply the area of a circle: $A(x) = \pi [f(x)]^2$.
    Example from notes: Proving the volume of a sphere is $\frac{4}{3}\pi r^3$ by rotating a semi-circle $y=\sqrt{r^2-x^2}$.
  • Washer Method: Use this when there is a gap between the axis of rotation and the solid. We subtract the inner hole from the outer solid. $$A(x) = \pi (R_{outer})^2 - \pi (r_{inner})^2$$

Watch out for shifted axes! As seen in the class notes (PDF 1, page 7), if we rotate around the line $y=2$ instead of the x-axis, our radius changes. We must calculate the distance from the axis of rotation to the curve (e.g., radius $= 2 - x^2$).

Section 6-3: The Cylindrical Shell Method

Sometimes, the Disk method is difficult because solving for a variable (like getting $x$ in terms of $y$) is messy or impossible. Enter the Shell Method. Instead of slicing perpendicular to the axis of rotation, we slice parallel to it, creating hollow cylinders nested inside each other like layers of an onion.

The formula for the volume using shells is based on the surface area of a cylinder:

$$V = \int_{a}^{b} 2\pi (\text{radius}) (\text{height}) \, dx$$
  • Radius ($r$): Usually just $x$ (distance from the y-axis), or $(x-c)$ if rotating around a different vertical line.
  • Height ($h$): The value of the function $f(x)$ (or top curve minus bottom curve).
  • Thickness: represented by $dx$ or $dy$.

Example from notes: Rotating the region bounded by $y=2x^2 - x^3$ about the y-axis. Using shells is much easier here because we don't have to solve for $x$. The integral becomes $V = \int_0^2 2\pi x (2x^2 - x^3) dx$, resulting in $\frac{16\pi}{5}$.

Comparison: Which Method Should I Use?

One of the most useful slides in the class notes (PDF 1, Page 15) compares the two methods directly. Here is the rule of thumb:

  1. Draw a typical rectangle in your region (perpendicular to the axis for Disks, parallel to the axis for Shells).
  2. Disk/Washer Method: The rectangle is perpendicular to the axis of rotation. You integrate with respect to the variable of the axis (e.g., rotate around x-axis $\rightarrow$ integrate $dx$).
  3. Cylindrical Shell Method: The rectangle is parallel to the axis of rotation. You integrate with respect to the opposite variable (e.g., rotate around y-axis $\rightarrow$ integrate $dx$).

Mastering both techniques allows you to choose the easiest path to the solution. Be sure to review the attached PDF notes for the step-by-step algebraic solutions to the specific homework problems covered in the video!