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Definite integrals compute net area.

The process of approximating areas under curves led to the notion of a Riemann sum

where $f$ is a nonnegative, continuous function on the interval $[a,b]$, and

$x_k^*$ is a sample point for the $k^{th}$ rectangle, $k=1,2,..., n$.

The limit of Riemann sums, as $n\to \infty$, gives the exact area between the curve $y=f(x)$ and the interval on the $x-$axis:

It does not matter whether we consider only right Riemann sums, or left Riemann sums, or midpoint Riemann sums, or others: the limit of any kind of Riemann sum as $n\to \infty$ is equal to the area, as long as $f$ is nonnegative and continuous on $[a,b]$.

What happens when a continuous function $f$ assumes negative values on the interval $[a,b]$?

We can still form Riemann sums and take the limit.

The question is: What is the meaning of a Riemann sum in that case?

When we take the limit of Riemann sums, it seems that we should get that

where the areas $A_1$ and $A_2$ are depicted in the figure below.

In other words, the limit of Riemann sums seems to be equal to the sum of signed areas of the regions that lie entirely above or below the $x$-axis. The signed area of regions that lie above the $x$-axis is positive, and the signed area of regions that lie below the $x$-axis is negative.

This sum of signed areas is called the net area of the region between the graph of $f$ and the interval on the $x$- axis .

The limit of Riemann sums will exist for any continuous functions on the interval $[a,b]$, even if $f$ assumes negative values on $[a,b]$. The limit of Riemann sums gives the net area of the region between the graph of $f$ and an interval on the $x$- axis.

This leads to the following definition.

The definite integral is a number that gives the net area of the region between the curve $y=f(x)$ and the $x$-axis on the interval $[a,b]$.

The definite integral computes the net area (sum of signed areas) between $y=f(x)$ and the $x$-axis on the interval $[a,b]$.

Consider the graph of a function $f$ on the interval $[0,5]$. Compute the definite integral
There is more to this example, than just a value of the integral.

Notice that and It follows that

This looks like a property of the definite integral. Are there other properties?

Due to the geometric nature of integration, geometric properties of functions can help us compute integrals.

The names odd and even come from the fact that these properties are shared by functions of the form $x^n$ where $n$ is either odd or even. For example, if $f(x) = x^3$, then and if $g(x) = x^4$, then Geometrically, even functions have symmetry with respect to the $y-$axis . Cosine is an even function:

On the other hand, odd functions have $180^\circ$ rotational symmetry around the origin. Sine is an odd function:
Let $f$ be an odd function defined for all real numbers. Compute:
Since our function is odd, it must look something like:
The integral above computes the following net area:
Let $f$ be an odd function defined for all real numbers. Which of the following are equal to
$\int _{4}^{2} f(x) \d x$ $\int _{-4}^{-2} f(x) \d x$ $\int _{-2}^{-4} f(x) \d x$ $\int _{-2}^{4} f(x) \d x$ $\int _{4}^{-2} f(x) \d x$ $\int _{2}^{-4} f(x) \d x$ $\int _{-4}^{2} f(x) \d x$ $-\int _{-4}^{2} f(x) \d x$ $-\int _{-4}^{-2} f(x) \d x$

### Net area versus geometric area

We know that the net area of the region between a curve $y=f(x)$ and the $x$-axis on $[a,b]$ is given by On the other hand, if we want to know the geometric area, meaning the “actual” area, we compute

True or false:
true false