Welcome back to class! In Section 6-3, we are expanding our toolkit for finding volumes of solids of revolution. While we previously used the Disk and Washer methods, today we introduce the Method of Cylindrical Shells. This method is often mathematically easier when the function is difficult to solve for a specific variable or when the region suggests a "vertical slice" approach for a vertical axis rotation.
The Core Formula
Instead of slicing the solid into flat disks, imagine the solid is made up of thin, nested cylinders—like the layers of an onion. To find the volume, we "unroll" these shells. The geometric formula relies on the circumference, the height of the shell, and the thickness.
The general volume integral using shells is:
$$V = \int_a^b 2\pi (\text{radius})(\text{height}) \, dx$$Key Takeaways from the Class Notes
- Rotation about the Y-Axis: When rotating a region bounded by $y=f(x)$ about the $y$-axis, the shell method is usually the most straightforward.
- The radius ($r$) is simply the distance from the axis of rotation to the slice, which is usually just $x$.
- The height ($h$) is the value of the function, $f(x)$.
- Example: For the region bounded by $y = 2x^2 - x^3$, the volume integral becomes $\int_{0}^{2} 2\pi x (2x^2 - x^3) \, dx$.
- Rotation about the X-Axis: To use shells for rotation about the $x$-axis, we must integrate with respect to $y$.
- Here, the radius is $y$ and the height is a function of $y$ (right curve minus left curve).
- Example: For $y=\sqrt{x}$ rotated about the $x$-axis, we solve for $x$ to get $x=y^2$. The height of the shell is $(1 - y^2)$, leading to the integral $\int_{0}^{1} 2\pi y (1-y^2) \, dy$.
Handling Shifted Axes
Things get interesting when we don't rotate around the standard axes. As seen in Example 4 from the notes, if we rotate around the line $x = 2$, our radius changes.
If the slice is at $x$ and the axis is at $x=2$, the distance (radius) is no longer just $x$; it becomes $2 - x$. Always draw a picture to verify your radius!
Disk vs. Shell: Which to Choose?
One of the most helpful tips from this lecture (see the notes on page 12) is knowing which method to pick based on your variable of integration:
- Disk/Washer Method: The cut is perpendicular to the axis of rotation.
- Rotate around X-axis $\rightarrow$ integrate $dx$.
- Rotate around Y-axis $\rightarrow$ integrate $dy$.
- Shell Method: The cut is parallel to the axis of rotation.
- Rotate around Y-axis $\rightarrow$ integrate $dx$ (often easier for $y=f(x)$ functions).
- Rotate around X-axis $\rightarrow$ integrate $dy$.
Remember, the goal is to set up an integral that is easy to evaluate. Sketching the region, identifying your $r$ and $h$, and picking the right method will ensure success. Happy integrating!