Definite integrals compute net area.
where is a nonnegative, continuous function on the interval , and
is a sample point for the rectangle, .
The limit of Riemann sums, as , gives the exact area between the curve and the interval on the 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 is equal to the area, as long as is nonnegative and continuous on .
What happens when a continuous function assumes negative values on the interval ?
We can still form Riemann sums and take the limit.
The question is: What is the meaning of a Riemann sum in that case?
The figure illustrates a right Riemann sum with rectangles for on .
So, the Riemann sum is the sum of signed areas of rectangles: rectangles that lie above the -axis contribute positive values, and rectangles that lie below the -axis contribute negative values to the Riemann sum.
When we take the limit of Riemann sums, it seems that we should get that
where the areas and 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 -axis. The signed area of regions that lie above the -axis is positive, and the signed area of regions that lie below the -axis is negative.
This sum of signed areas is called the net area of the region between the graph of and the interval on the - axis .
The limit of Riemann sums will exist for any continuous functions on the interval , even if assumes negative values on . The limit of Riemann sums gives the net area of the region between the graph of and an interval on the - axis.
This leads to the following definition.
- by interpreting the integral as the net area of the region between the curve and the interval on the -axis;
- using the definition of the definite integral, i.e. by computing the limit of Riemann sums.
- The area between the -axis and the curve can be easily computed, since it is the
area of a triangle.
Then, it follows that
- We use the definition of the definite integral and write It does not matter
what type of a Riemann sum we use, so we choose a right Riemann
A right Riemann sum with is illustrated in the figure below.
We can apply the constant multiple rule for sums and limits:
We can now finish our computation of the limit of right Riemann sums.
- Express the limit as a definite integral.
- Compute this limit:
The definite integral computes the net area (sum of signed areas) between and the -axis on the interval .
Notice that and It follows that
This looks like a property of the definite integral. Are there other properties?
- Here, there is no “area under the curve” when the region has no width; hence this definite integral is .
- This states that total area is the sum of the areas of subregions. Here a picture
is worth a thousand words:
- For now, this property can be viewed as merely a convention to make other properties work well. However, later we will see how this property has a justification all its own.
- This states that when one scales a function by, for instance, , the area of the enclosed region also is scaled by a factor of .
- This states that the integral of the sum is the sum of the integrals.
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 where is either odd or even. For example, if , then and if , then Geometrically, even functions have symmetry with respect to the axis . Cosine is an even function:
We know that the net area of the region between a curve and the -axis on is given by On the other hand, if we want to know the geometric area, meaning the “actual” area, we compute
- Express the geometric area of the region between the curve and the -axis on the interval as a definite integral.
- Express the geometric area of the region between the curve and the -axis on the interval in terms of definite integrals of .
- Express the geometric area of the region between the curve and the -axis on the interval in terms of areas , , and .