Integral calculus

Integral calculus deals with the properties and applications of integrals. Physically, the integral of a function $f(x)$, written $\int f(x)\,dx$, represents the AREA under the curve of $f(x)$ plotted against $x$. Integration is the reverse process of differentiation.

Motivating the definite integral. For simple shapes (a rectangle, a triangle) the area under a straight-line graph between $x=a$ and $x=b$ can be found directly — e.g. as the sum of a rectangle of area $f(a)(b-a)$ and a triangle of area $\tfrac12(b-a)\big(f(b)-f(a)\big)$. For a general curve, no such simple formula exists, so the area is approximated by dividing the region into a large number $n$ of thin vertical strips, each treated as a rectangle, and summing their areas:

As $n\to\infty$ (each strip becomes infinitesimally thin), this sum converges to the EXACT area under the curve:

This limiting sum, from $x=a$ to $x=b$, is called the definite integral of $f(x)$ and is written

Indefinite integral and the link to differentiation. $F(x) = \int f(x)\,dx$ --- (2.42) is called the indefinite integral of $f(x)$ (no limits on $x$). Since integration reverses differentiation, $f(x) = \dfrac{d}{dx}\big(F(x)\big)$ --- (2.43), and the definite integral is evaluated as

Properties of integration.

1. $\int\big(f_1(x)+f_2(x)\big)\,dx = \int f_1(x)\,dx+\int f_2(x)\,dx$ --- (2.45)
2. $\int K f(x)\,dx = K\int f(x)\,dx$, for $K=$ constant --- (2.46)

Standard indefinite integrals.

$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} \quad \text{--- (2.47)} \qquad \int \dfrac1x\,dx = \ln x \quad \text{--- (2.48)}$ …