AcademyIntegration
Academy
Definite Integrals
Level 1 - Math I (Physics) topic page in Integration.
Definite Integrals
A definite integral has upper and lower limits (bounds) and computes a specific numerical value, typically representing the signed area under a curve.
Definition
Definite Integral
\[\int_a^b f(x) \, dx = F(b) - F(a)\]
where \(F(x)\) is any antiderivative of \(f(x)\). This is the Evaluation Theorem, also called the Second Fundamental Theorem of Calculus.
Area Under a Curve
When \(f(x) \geq 0\) on \([a,b]\), the definite integral represents the area between the curve and the x-axis:
Area
\[\text{Area} = \int_a^b f(x) \, dx \quad \text{when } f(x) \geq 0\]
Properties of Definite Integrals
Property 1
\[\int_a^a f(x) \, dx = 0\]
Property 2
\[\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx\]
Property 3
\[\int_a^b [f(x) + g(x)] \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx\]
Property 4
\[\int_a^b cf(x) \, dx = c \int_a^b f(x) \, dx \quad \text{for constant } c\]
Example
Evaluate \(\int_0^2 x^2 \, dx\):
Example
\[\int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} - 0 = \frac{8}{3}\]
Area Between Curves
The area between two curves \(f(x)\) and \(g(x)\) from \(a\) to \(b\):
Between Curves
\[\text{Area} = \int_a^b |f(x) - g(x)| \, dx\]
When \(f(x) \geq g(x)\) on \([a,b]\): \(\int_a^b [f(x) - g(x)] \, dx\)