Definite integrals

Let $F:I\to \mathbb{R}$ be an antiderivative of the continuous function $f:I\to \mathbb{R}$. The definite integral of f from a to b is the real number:

(the Leibniz–Newton formula)

Properties:

• Linearity of the integral:

• Additivity of the integral:

• The mean value property of the integral: if $f\ge0$ on

• Monotonicity of the integral: if $f\le g$ on

Geometric interpretation of the definite integral

Let a<b be real numbers and $f:[a,b]\to \mathbb{R}$ a positive continuous function. The region under the graph of f is

The Riemann sum (or integral sum) associated with f, $\Delta$ and $\xi$ is the real number where with , $i\in \{1,2,...,n\}$ being the system of intermediate points associated with the partition: .

The area under the graph of a positive continuous function $f:[a,b]\to \mathbb{R}$ is: area .

The Riemann integral of piecewise continuous functions

A function $f:[a,b]\to \mathbb{R}$ is called piecewise continuous if it has at most a finite, non-zero number of points of discontinuity, and these are discontinuities of the first kind.

Integrating piecewise continuous functions. A piecewise continuous function $f:[a,b]\to \mathbb{R}$ is integrable on $[a,b]$ and , where are the continuous functions associated with f.

Methods for computing definite integrals

Integration by parts. Suppose the functions $f,g:I\to \mathbb{R}$ are differentiable, with continuous derivatives . Let $a,b\in I$ be two numbers. Then:

Change of variable. Suppose the function $ϕ:J\to I$ is differentiable with continuous derivative, and the function $F:I\to \mathbb{R}$ is continuous. Let $\alpha,\beta \in J$ be two numbers. Then: One changes both variable and symbol, $ϕ(t)=x$ and $ϕ'(t)dt=dx, t\in J$ and $x\in I$.

Further properties:

The mean value property of the integral

For the function $f$, there exists a number $c\in(a,b)$ such that

The mean value inequality

If $m,M\in \mathbb{R}$ are such that $m\le f(x)\le M$, $\forall x\in[a,b]$, then

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