Cylindrical Shells
Sometimes it is easier to rotate around the y-axis. Instead of discs, we use Cylindrical Shells. Each shell has a circumference \(2\pi x\), a height \(f(x)\), and a thickness \(dx\).
\[V = \int_a^b 2\pi x f(x) dx\]
Worked Examples
Example 1: Rotating around the Y-axis
Find the volume when \(y = x^2\) from 0 to 1 is rotated around the y-axis.
- Integral: \(\int_0^1 2\pi x (x^2) dx = 2\pi \int_0^1 x^3 dx = 2\pi [\frac{x^4}{4}]_0^1 = \frac{\pi}{2}\).
- Result: \(\frac{\pi}{2}\).
The Bridge to Quantum Mechanics
The Shell Method is mathematically identical to calculating the Charge Distribution in a spherical atom. When we say an electron is "in a 1s orbital," we are describing a spherical shell of probability. The formula \(P(r) dr = 4\pi r^2 |\psi(r)|^2 dr\) is simply the shell method applied to a sphere. It tells us that even though the wavefunction is strongest at the center, the electron is most likely to be found at a certain "shell" distance away.