1 Definite integrals

We saw in the previous Section that $\int \; <!--nolimits\; -->f\left(x\right)dx=F\left(x\right)+c$ where $F\left(x\right)$ is that function which, when differentiated, gives $f\left(x\right)$ . That is, $\frac{dF}{dx}=f\left(x\right)$ . For example,

$\int \; <!--nolimits\; -->sin\left(3x\right)dx=-\frac{cos\left(3x\right)}{3}+c$

Here, $f\left(x\right)=sin\left(3x\right)$ and $F\left(x\right)=-\frac{1}{3}cos\left(3x\right)$ We now consider a definite integral which is simply an indefinite integral but with numbers written to the upper and lower right of the integral sign. The quantity

${\int \; <!--nolimits\; -->}_a^{b}f\left(x\right)dx$

is called the definite integral of $f\left(x\right)$ from $a$ to $b$ . The numbers $a$ and $b$ are known as the lower limit and upper limit respectively of the integral. We define

${\int \; <!--nolimits\; -->}_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$

so that a definite integral is usually a number. The meaning of a definite integral will be developed in later Sections. For the present we concentrate on the process of evaluating definite integrals.