Calculus Essentials: Definite Integrals

Quantum mechanics measures probabilities, and a probability is an area under a curve — the area under $|\psi(x)|^2$. Computing those areas is what the definite integral does. This lesson gives you exactly the calculus you need: how to evaluate $\int_a^b f(x)\,dx$ for the polynomial integrands that show up in normalization, expectation values, and probability over an interval.

The definite integral as area

For a function $f(x)$ on an interval $[a, b]$, the definite integral

is the (signed) area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$. When $f(x) = |\psi(x)|^2$ is a probability density, that area is the probability of finding the particle between $a$ and $b$.

The power rule

An antiderivative $F(x)$ of $f(x)$ is any function whose derivative is $f$: $F'(x) = f(x)$. For powers of $x$, the antiderivative follows one rule — raise the exponent by one and divide by the new exponent:

A sum integrates term by term, and a constant multiple pulls outside the integral, so any polynomial integrates immediately. For example $\int (x^2 - 2x^3)\,dx = \tfrac{x^3}{3} - \tfrac{x^4}{2}$.

The Fundamental Theorem of Calculus

To turn an antiderivative into a definite integral, evaluate it at the two endpoints and subtract. This is the Fundamental Theorem of Calculus: if $F'(x) = f(x)$, then

The notation $F(x)\big|_a^b$ is just shorthand for "plug in $b$, then subtract the value at $a$."

A worked example

Evaluate $\displaystyle\int_0^1 x^2(1-x)^2\,dx$ — the exact integral you will meet again when normalizing a wavefunction. First expand the integrand:

Integrate term by term with the power rule:

At $x = 1$ every power is $1$ and at $x = 0$ every term vanishes, so by the Fundamental Theorem:

Try it

This is a numerical exercise — your code should return a number. Compute the definite integral $\displaystyle\int_0^1 x^2(1-x)^2\,dx$ and return its value.

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