Area Under a Curve
To find the exact area under a curve \(f(x)\), we can divide the region into \(n\) rectangles. As \(n\) goes to infinity, the width of each rectangle goes to zero, and the sum of their areas becomes the exact area. This is a Riemann Sum.
\[A = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x\]
Worked Examples
Example 1: Visualizing Summation
Imagine finding the area under \(y = x\) from 0 to 1. If you use 2 rectangles, you get a rough estimate. If you use 1,000, you get a very close estimate. If you use an infinite number, you get exactly \(0.5\) (the area of a triangle).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we don't just calculate areas; we calculate Expectation Values. These are essentially weighted sums of all possible measurement outcomes. Just as a Riemann Sum adds up tiny rectangles to find a total area, the Path Integral adds up all possible paths a particle can take to find the total probability of it moving from A to B. Summing over the "infinitesimal" is the only way to capture the full quantum nature of reality.