Definite Integrals

Definite integrals are one of the most important ideas in calculus. They allow us to calculate area, total change, distance, accumulated quantity, and much more.

A definite integral looks like this:

$$\int_{a}^{b}f(x)dx$$

It represents the net area between the graph of f(x) and the x-axis from x=a to x=b

Fundamental Theorem of Calculus (FTC)

The FTC is what actually allows us to compute definite integrals easily.

✔ Part 1 — Derivative of an integral

If

$$F(x)=\int_{a}^{x}f(t)dt$$

then

$$F^\prime(x)=f(x)$$

→ Differentiation and integration are inverse processes.

✔ Part 2 — Evaluating a definite integral

If $$f^\prime(x)=f(x)$$, then:

$$\int_{a}^{b}f(x)dx=F(b)-F(a)$$

This is the most practical formula for calculations:

👉 Definite integral = antiderivative evaluated at endpoints

Basic Properties of Definite Integrals

1. Reversing limits changes sign

$$\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx$$

2. If both limits are the same

$$\int_{a}^{a}f(x)dx=0$$

3. Additivity

$$\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx$$

4. Constant multiple rule

$$\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx$$

5. Comparison

If $$f(x) \geq g(x)$$,

$$\int_{a}^{b}f(x)dx \geq \int_{a}^{b}g(x)dx$$

Example 1: Power Function

$$\int_{0}^{3}x^{2}dx$$

$$=\frac{1}{3}x^{3}\big|_{0}^{3} = \frac{1}{3}3^{2}-\frac{1}{3}0=\frac{1}{3}\cdot27=9$$

Example 2: Area Below x-axis

$$\int_{-1}^{1}-3xdx$$

$$-\frac{3}{2}x^{2}\big|_{-1}^{1}=\left(-\frac{3}{2}(1)^{2}\right)-\left(-\frac{3}{2}(-1)^{2}\right) = 0$$

odd functions “perfect symmetry” integrals are cancels out.

Example 3: Trigonometric

$$\int_{0}^{\pi}\sin(x)dx$$

$$=-\cos(x)\big|_{0}^{\pi}=(-\cos(\pi)-(-\cos(0)_)=-(-1)-(-1)=2$$