310 Definite Integral
A scalar defined as the limit of a sum of function values on subintervals that is used to measure accumulated size along an interval.
definition [d] (Definite Integral) The number obtained from a function \(f\) on \([a, b]\) by partitioning \([a, b]\) into \(n\) subintervals of equal width \(\Delta x = (b-a)/n\), choosing sample points \(x_{i}^{*}\) in each, and taking the limit of Riemann sums:
- \(\displaystyle \int_{a}^{b} f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i}^{*})\, \Delta x\) ,
provided the limit exists and is independent of the choice of sample points.
where
- \(a\) is the lower limit; \(b\) is the upper limit.
- \(\Delta x = (b-a)/n\) is the subinterval width.
- \(x_{i}^{*}\) is a sample point in the \(i\)th subinterval \([x_{i-1},\, x_{i}]\).
- \(f\) is integrable on \([a,b]\) when this limit exists.
310.1 References
- Stewart, J. Calculus. — definite integral as a limit of Riemann sums.
- Basic Derivation of Jacobian
- Chain Rule
- Cross Product
- Curl
- Curve
- Definite Integral
- Derivation of Curl
- Derivation of Divergence
- Derivation of Grad
- Derivation of Jacobian
- Derivation of Taylor Expansion
- Divergence
- Dot Product
- Envelope of Tangent Lines
- General Functions
- Gradient
- Improper Integral
- Indefinite Integral
- Jacobian
- Legendre Transform
- Legendre Transform Derivation
- Limit
- Line Integral
- Notes
- Parameterization
- Partial Derivative
- Real Parameter
- Scalar Projection
- Smooth Curve
- Vector Calculus
- Vector Differential Operator
- Vector Field
- Vector Projection