The Difference of Sums
To find the area between two curves \(f(x)\) and \(g(x)\), we simply subtract the lower curve from the upper curve and integrate:
\[A = \int_a^b [f(x) - g(x)] dx\]
Worked Examples
Example 1: Parabola and Line
Find the area between \(y = x^2\) and \(y = x\).
- Intersection points: \(x^2 = x \implies x = 0, 1\).
- Upper curve on \((0, 1)\) is \(y = x\).
- Integral: \(\int_0^1 (x - x^2) dx = [\frac{x^2}{2} - \frac{x^3}{3}]_0^1 = 0.5 - 0.33 = 1/6\).
- Result: \(1/6\).
The Bridge to Quantum Mechanics
Area between curves is used to calculate Transition Probabilities. When an electron jumps between two energy levels, the probability of that jump depends on the "overlap" between the two wavefunctions. This overlap is calculated by integrating the product of the two functions, which is geometrically related to the area they share. This is what determines why some transitions are "allowed" (bright lines in a spectrum) and others are "forbidden" (dark lines).