Definite integrals

Summary

• Throughout the section, $f$ is a continuous function defined on $I=[a,b]$ with $a<b$ .
• If $f(x)≥0$ for all $x∈I$ then we define

$$\int_{a}^{b}{ ‍ }f(x)dx$$

• as the area of the shape bounded by the graph of the function, the $x$ -axis and the vertical lines through $x=a$ and $x=b$ .

• Notation:
• The symbol $\int{ ‍ }$ is called the integral sign.
• The function inside the integral between the integral sign and the symbol $dx$ is called the integrand .
• $a,b$ are called the (lower and upper) limits of the integral.
• $dx$ indicates that $x$ is the variable of integration.
• $\int_{a}^{b}{ ‍ }f(x)dx$ is always a real number.
• $x$ is a dummy-variable and $\int_{a}^{b}{ ‍ }f(x)dx$ = $\int_{a}^{b}{ ‍ }f(u)du$ .
• If $f(x)≤0$ for all $x∈I$ then we define

$$\int_{a}^{b}{ ‍ }f(x)dx$$

• as minus the area of the shape bounded by the graph, the $x$ -axis, $x=a$ and $x=b$ .
• In general, when $f\left( x \right)$ takes both positive and negative values, we define $\int_{a}^{b}{ ‍ }f(x)dx$ as the area above the $x$ -axis minus the area below the $x$ -axis.
• For convenience, we define

$$\int_{b}^{a}{ ‍ }f(x)dx=-\int_{a}^{b}{ ‍ }f(x)dx$$

• and for any $c∈(a,b)$ we define

$$\int_{c}^{c}{ ‍ }f(x)dx=0$$

• This is not how $\int_{a}^{b}{ ‍ }f(x)dx$ is defined formally . Actually, there are several different definitions of $\int_{a}^{b}{ ‍ }f(x)dx$ . However, with the given assumptions, all definitions agree and they also agree with our informal understanding of area. Also, continuity is a sufficient but not necessary for the integral to exist.