Total Sum
A Definite Integral has upper and lower limits. It represents the "Total Accumulation" of a quantity between two points.
\[\int_a^b f(x) dx\]
Unlike the indefinite integral, the result is a number, not a function.
Worked Examples
Example 1: Displacement from Velocity
If a car's velocity is \(v(t) = 2t\), how far does it travel between \(t=0\) and \(t=3\)?
- The distance is the area under the velocity curve: \(\int_0^3 2t dt\).
- Antiderivative is \(t^2\).
- Evaluate at limits: \(3^2 - 0^2 = 9\).
- Result: 9 meters.
The Bridge to Quantum Mechanics
In Quantum Mechanics, we use definite integrals to calculate the Total Probability of finding a particle in a region. Since the particle must exist somewhere, the integral of its probability density over all space must be exactly 1:
\[\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1\]
This is called Normalization. If this integral isn't 1, the math of Quantum Mechanics breaks down—you can't have a 110% chance of a particle existing!