Defining Integration: Riemann Sums and Signed Area

1Introduction

This lecture defines the definite integral as the limit of sums formed by adding the contribution from each subinterval. It also explains why a definite integral represents signed area and clarifies how signed area differs from geometric area.

2Definition

Let $a<b$a<b and consider a function $f:[a,b]\to \mathbb{R}$f:[a,b]\to\mathbb R. Partition $[a,b]$[a,b] as

and choose a sample point ${\xi }_i$\xi_i in each subinterval $[x_{i-1},x_i]$[x_{i-1},x_i]. The corresponding Riemann sum is

For a partition $P$P, define its mesh by

If the Riemann sums converge to the same limit as $\left|P\right|\to 0$|P|\to0, independently of both the partitions and the sample points, write this limit as

and say that $f$f is Riemann integrable on $[a,b]$[a,b].

3Integrability of Continuous Functions

3.1Theorem

Every function that is continuous on a closed interval $[a,b]$[a,b] is Riemann integrable on $[a,b]$[a,b].

3.2Proof

For a partition $P$P, let $M_i$M_i and $m_i$m_i be the maximum and minimum of $f$f on its $i$ith subinterval, and define

The lower integral $\underset{P}{\sup }L(f,P)$\sup_P L(f,P) does not exceed the upper integral $\underset{P}{\inf }U(f,P)$\inf_P U(f,P). If, for every $\epsilon >0$\varepsilon>0, some partition satisfies $U(f,P)-L(f,P)<\epsilon$U(f,P)-L(f,P)<\varepsilon, these two quantities have a common value $I$I.

To connect this criterion to tagged sums, take any tagged partition $Q$Q and its common refinement $R$R with $P$P. Since $R$R refines $P$P,

so every tagged sum on $R$R differs from $I$I by less than $\epsilon$\varepsilon. Let $\left|f\right|\text{[PARSE ERROR: Undefined("Command("le")")]}B$|f|\le B, and let $k$k be the number of interior division points of $P$P. Only the subintervals of $Q$Q containing these points change when $Q$Q is refined to $R$R; hence the corresponding tagged sums differ by at most $2Bk\left|Q\right|$2Bk|Q|. Therefore every tagged sum on $Q$Q tends to $I$I as $\left|Q\right|\to 0$|Q|\to0. This is the Darboux criterion.

A continuous function on a closed interval is uniformly continuous. Thus, when $\left|P\right|$|P| is sufficiently small, $M_i-m_i<\epsilon /(b-a)$M_i-m_i<\varepsilon/(b-a) on every subinterval. Consequently,

The Darboux criterion now shows that $f$f is Riemann integrable.

This theorem justifies the use of definite integrals of continuous functions in the subsequent fundamental theorem of calculus.

4Signed Area

In a definite integral, contributions above the $x$x-axis are positive and contributions below it are negative. Geometric area, which is always nonnegative, is calculated by integrating the absolute value $\left|f\right|$|f|. If $f$f has only finitely many sign-change points, one may instead split the interval at those points.

5Example

The function $f\left(x\right)=x$f(x)=x is odd, so its positive and negative contributions cancel on $[-1,1]$[-1,1]:

The geometric area is instead

Thus, the value of a definite integral and geometric area do not agree in general.

6Exercises

7Related Material