Lecture Note: §5 Integration

Published: November 11, 2025

Last Update: 2025-11-19

The Sigma notation: \(\sum_{k=1}^{n} a_{k} = a_{1} + a_{2} + a_{3} + \cdots + a_{n-1} + a_{n}\).
The Riemann Sums for \(f\) on the interval \([a, b]\)
- Partition and its norm.

Definite Integral
- \(\int_{a}^{b} f(x) ,dx = \lim_{}\sum\limits_{i=1}^{n} f(\xi_i)\Delta x_i\)
- If \(f(x) \geq 0\), the area under the curve \(f(x)\) over \([a, b]\):
\[A = \int_{a}^{b} f(x)\, dx.\]
- \(\text{av}(f) = \frac{1}{b-a}\int_{a}^{b} f(x) dx\)
Integrability of Continuous Functions
Rules satisfied by definite integrals

The Mean Value Theorem for Definite Integrals
- If \(f\) is continuous on \([a,b]\), then at some point \(c\) in \([a,b]\), \[f(c) = \frac{1}{b-a}\int_{a}^{b}f(x)\,dx.\]
Define a New function by Integrals
- If \(f\) is an integrable function, one can define a function \(F\) whose value at \(x\) is \(F(x) = \int_{a}^{x} f(t)\,dt.\)
The Fundamental Theorem of Calculus
- If \(f(x)\) is continuous on \([a,b]\), then \(F(x) = \int_{a}^{x} f(t) \,dt\) is continuous on \([a, b]\) and differentiable on \((a,b)\) and its derivative is \(f(x)\):
\[ F’(x) = \frac{d}{dx}\int_{a}^{x} f(t) \, dt = f(x).\]
- If \(f(x)\) is continuous on \([a, b]\) and \(F(x)\) is any antiderivative of \(f\) on \([a, b]\), then
\[ \int_{a}^{b} f(x) \,dx = F(b) - F(a).\]
The Relationship Between Integration and Differentiation
Total Area: \(A = \int_{a}^{b} \lvert f(x) \rvert\,dx\).

The Substitution Rule \(\int f(g(x))g’(x) dx = \int f(u) du\) with \(u = g(x)\).
- Running the chain rule backwards.
- The general version, the change of variable.

Substitution in Definite Integrals
- \(\int_{a}^{b} f(g(x))g’(x) dx = \int_{g(a)}^{g(b)} f(u)du\) with \(u = g(x)\)
The Definite Integrals of Even or Odd functions on the symmetric interval \([-a, a]\).
- If \(f\) is odd, \(\int_{-a}^{a} f(x) \,dx = 0\).
- If \(f\) is even, \(\int_{-a}^{a} f(x) \,dx = 2\int_{0}^{a} f(x)\,dx\).
Areas Between Curves: \(A = \int_{a}^{b} \lvert f(x) - g(x) \rvert\,dx\).