Calculating Area Easily
Part II of the FTC gives us a practical way to calculate definite integrals. Instead of adding infinite rectangles, we just find the antiderivative and subtract the values at the endpoints:
\[\int_a^b f(x) dx = F(b) - F(a)\]
where \(F\) is any antiderivative of \(f\).
Worked Examples
Example 1: Basic Area
Evaluate \(\int_1^3 x^2 dx\).
- Antiderivative: \(F(x) = \frac{x^3}{3}\).
- Evaluate: \(F(3) - F(1) = \frac{27}{3} - \frac{1}{3} = 9 - 0.33 = 8.67\).
- Result: \(\frac{26}{3} \approx 8.67\).
Example 2: Exponential Accumulation
Evaluate \(\int_0^{\ln 2} e^x dx\).
- Antiderivative: \(F(x) = e^x\).
- Evaluate: \(F(\ln 2) - F(0) = e^{\ln 2} - e^0 = 2 - 1 = 1\).
- Result: 1.
The Bridge to Quantum Mechanics
We use FTC Part II to calculate the Expectation Value of Position \(\langle x \rangle = \int x |\psi(x)|^2 dx\). This integral tells us where a particle is, on average. By using the antiderivative method, we can quickly solve for the behavior of particles in simple systems like the "Particle in a Box." The FTC transforms a difficult summation over infinite points into a simple subtraction of two numbers.