1. What a Definite Integral Means
A definite integral gives a numerical value. It measures the net area between the graph of a function and the x-axis over a specific interval. In simple notes-style: an indefinite integral gives a family of functions, but a definite integral gives a single number.
2. Notation of Definite Integral
A definite integral uses upper and lower limits on the integral symbol.
2.1. General Form
\int_a^b f(x) \, dx
This represents the signed area from x = a to x = b.
3. Signed Area Interpretation
A definite integral does not always give geometric area, but 'signed' area:
- Area above the x-axis is positive.
- Area below the x-axis is negative.
This helps track how the function moves relative to the axis.
4. Connection Between Definite and Indefinite Integrals
A definite integral is evaluated using antiderivatives. This shortcut comes from the Fundamental Theorem of Calculus.
4.1. Evaluation Rule
\int_a^b f(x) \, dx = F(b) - F(a)
\text{where } F'(x) = f(x)
4.2. Meaning
You do not integrate with the limits attached. First find the antiderivative F(x), then plug in the limits b and a.
5. Basic Properties of Definite Integrals
These simple properties help recognize and simplify integrals quickly.
5.1. Properties
\int_a^a f(x) dx = 0
\int_a^b f(x) dx = -\int_b^a f(x) dx
\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx
6. Examples
These examples show the basic process of evaluating definite integrals using antiderivatives.
6.1. Example 1 — Polynomial
Evaluate:
\int_0^2 (3x^2) \, dx
Antiderivative:
F(x) = x^3
Apply limits:
F(2) - F(0) = 8 - 0 = 8
6.2. Example 2 — Trigonometric
Evaluate:
\int_0^{\pi} \sin x \, dx
Antiderivative:
F(x) = -\cos x
Apply limits:
[-\cos x]_0^{\pi} = [-(-1)] - [-(1)] = 2
6.3. Example 3 — Exponential
Evaluate:
\int_0^1 e^x dx
Antiderivative:
F(x) = e^x
Apply limits:
e - 1
7. Why Definite Integrals Matter
Definite integrals are used to compute areas, distances, total quantities, averages, and many real-life measurements. They form the backbone of most applications of calculus.