Guide to Applying Disk, Washer, and Shell Methods Based on the Problem

📚 Understanding the Disk Method

The disk method is used to find the volume of a solid of revolution when you rotate a region around an axis, and the slices perpendicular to the axis of rotation are solid disks. This method is best applied when there is no gap between the function and the axis of rotation.

• 📏 Definition: The disk method calculates volume by summing the volumes of infinitesimally thin disks.
• 📜 History: The concept originates from integral calculus, developed in the 17th century by Newton and Leibniz.
• 🔑 Key Principle: If $r(x)$ is the radius of the disk at position $x$, the volume $V$ is given by $V = \pi \int_a^b [r(x)]^2 dx$ for rotation about the x-axis, or $V = \pi \int_c^d [r(y)]^2 dy$ for rotation about the y-axis.
• 🌍 Real-world Example: Imagine rotating the region under the curve $y = \sqrt{x}$ from $x = 0$ to $x = 4$ about the x-axis. The radius is $r(x) = \sqrt{x}$, so the volume is $V = \pi \int_0^4 (\sqrt{x})^2 dx = \pi \int_0^4 x dx = 8\pi$.
• 💡 Conclusion: Use the disk method when rotating a region directly adjacent to the axis of rotation.

📚 Understanding the Washer Method

The washer method is an extension of the disk method used when the region being rotated has a gap between it and the axis of rotation. This creates a “washer” shape (a disk with a hole in the center).

• 📏 Definition: The washer method calculates volume by subtracting the volume of the “hole” from the volume of the outer disk.
• 📜 History: Also derived from integral calculus, expanding on the disk method for more complex shapes.
• 🔑 Key Principle: If $R(x)$ is the outer radius and $r(x)$ is the inner radius, the volume $V$ is given by $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) dx$ for rotation about the x-axis, or $V = \pi \int_c^d ([R(y)]^2 - [r(y)]^2) dy$ for rotation about the y-axis.
• 🌍 Real-world Example: Consider the region bounded by $y = x^2$ and $y = x$ rotated about the x-axis. Here, $R(x) = x$ and $r(x) = x^2$, so $V = \pi \int_0^1 (x^2 - x^4) dx = \frac{2\pi}{15}$.
• 💡 Conclusion: Use the washer method when rotating a region that does NOT directly touch the axis of rotation.

📚 Understanding the Shell Method

The shell method is used to find the volume of a solid of revolution by integrating along an axis that is parallel to the axis of rotation. This method is particularly useful when the function is difficult to express in terms of the other variable or when the axis of rotation makes the disk or washer method complicated.

• 📏 Definition: The shell method calculates volume by summing the volumes of infinitesimally thin cylindrical shells.
• 📜 History: An alternative approach to volume calculation, offering advantages in specific scenarios.
• 🔑 Key Principle: If $r(x)$ is the radius of the shell and $h(x)$ is the height, the volume $V$ is given by $V = 2\pi \int_a^b r(x)h(x) dx$ for vertical shells (rotation about the y-axis), or $V = 2\pi \int_c^d r(y)h(y) dy$ for horizontal shells (rotation about the x-axis).
• 🌍 Real-world Example: Rotate the region bounded by $y = x - x^2$ and the x-axis about the y-axis. Here, $r(x) = x$ and $h(x) = x - x^2$, so $V = 2\pi \int_0^1 x(x - x^2) dx = \frac{\pi}{6}$.
• 💡 Conclusion: Use the shell method when integrating parallel to the axis of rotation or when the disk/washer method is too complex.

📚 Choosing the Right Method: A Practical Guide

Here's a simple guide to help you decide which method to use:

📚 Practice Quiz

Test your understanding with these practice problems:

• ❓ Determine the volume of the solid formed by rotating the region bounded by $y = x^3$, $y = 0$, and $x = 1$ about the x-axis.
• ❓ Find the volume of the solid formed by rotating the region bounded by $y = x^2$ and $y = 4$ about the y-axis.
• ❓ Calculate the volume of the solid generated by rotating the region bounded by $y = \sqrt{x}$, $x = 4$, and $y = 0$ about the y-axis.