Crash introduction to infinitesimal calculus.
- Limits and Derivatives
- Integrals - this file
- Ordinary Differential Equations
See: Matematica: Calcolo infinitesimale: Integrali
Definite integrals are first introduced with its geometrical interpretation. The formal definition as the limit of a Riemann sum follows. Fundamental theorem of calculus provides the relation between integral and derivative.
Geometrical Interpretation
The value of the definite integral
\[\int_{x=a}^{b} f(x) \, dx \ ,\]
equals the area (with sign) in the Cartesian plane between the \(x\)-axis and the graph of the function \(y = f(x)\), and the vertical lines with coordinates \(x = a\), and \(x = b\). This value can be approximated with the sum of the area of rectangles or trapezoids (see APPROXIMATION RULEs): as the number of the elementary geometrical shapes of the approximation increases, and their height (in \(x\) direction) decreases, the approaximation becomes better and better. As the number of elements goes to \(\infty\), and the maximum height of the elementary shapes goes to zero, the definition of the definite Riemann integral follows
\[\int_{x=a}^{b} f(x) \, dx = \lim_{\max h_k \rightarrow 0} \sum_{k} \text{Area}_k \ .\]
If this limit exists, then the integral of the function \(f(x)\) between \(x=a\) and \(x=b\) exists. This definition is independent from the approximation (as long as it’s a definite approximation…): you can easily check that the value of the Riemann Sum Approximation tends to the value of the Exact Definite Integral, while the absolute error and the relative error percentage go to zero.