3D Shapes from 2D Curves
If you rotate a curve \(f(x)\) around the x-axis, you create a solid of revolution. We can find its volume by summing up infinitely thin "discs." Each disc has an area \(\pi [f(x)]^2\) and a thickness \(dx\).
\[V = \int_a^b \pi [f(x)]^2 dx\]
Worked Examples
Example 1: Volume of a Cone
Rotate the line \(y = x\) from 0 to 2 around the x-axis.
- Integral: \(\int_0^2 \pi (x)^2 dx = \pi [\frac{x^3}{3}]_0^2 = \frac{8\pi}{3}\).
- Result: \(\frac{8\pi}{3}\).
The Bridge to Quantum Mechanics
Atoms are 3D objects. When we calculate the Radial Probability Density of an electron, we are essentially looking at the volume of a spherical shell. The "Disc Method" is the first step toward understanding how a 1D wavefunction \(\psi(r)\) represents a 3D cloud of probability. It is how we calculate the total "amount" of electron charge within a certain distance from the nucleus.