## Learning Objectives

A student will be able to:

- Compute the area between two curves with respect to the and axes.

In the last chapter, we introduced the definite integral to find the area between a curve and the axis over an interval In this lesson, we will show how to calculate the area between two curves.

Consider the region bounded by the graphs and between and as shown in the figures below. If the two graphs lie above the axis, we can interpret the area that is sandwiched between them as the area under the graph of subtracted from the area under the graph

Therefore, as the graphs show, it makes sense to say that

[Area under (Fig. 1a)] [Area under (Fig. 1b)] [Area between and (Fig. 1c)],

This relation is valid as long as the two functions are continuous and the upper function on the interval

**The Area Between Two Curves** (*With respect to the* *axis*)

If and are two continuous functions on the interval and for all values of in the interval, then the area of the region that is bounded by the two functions is given by

**Example 1**:

Find the area of the region enclosed between and

**Solution:**

We first make a sketch of the region (Figure 2) and find the end points of the region. To do so, we simply equate the two functions,

and then solve for

from which we get and

So the upper and lower boundaries intersect at points and

As you can see from the graph, and hence and in the interval Applying the area formula,

Integrating,

So the area between the two curves and is

Sometimes it is possible to apply the area formula with respect to the coordinates instead of the coordinates. In this case, the equations of the boundaries will be written in such a way that is expressed explicitly as a function of (Figure 3).

**The Area Between Two Curves** (*With respect to the axis*)

If and are two continuous functions on the interval and for all values of in the interval, then the area of the region that is bounded by on the left, on the right, below by and above by is given by

**Example 2:**

Find the area of the region enclosed by and

**Solution:**

As you can see from Figure 4, the left boundary is and the right boundary is The region extends over the interval However, we must express the equations in terms of We rewrite

Thus

## Multimedia Links

For a video presentation of the area between two graphs **(14.0)(16.0)**, see Math Video Tutorials by James Sousa, Area Between Two Graphs (6:12).

For an additional video presentation of the area between two curves **(14.0)(16.0)**, see Just Math Tutoring, Finding Areas Between Curves (9:51).

## Review Questions

In problems #1 - 7, sketch the region enclosed by the curves and find the area.

- on the interval
- on the interval
- integrate with respect to
- integrate with respect to
- Find the area enclosed by and
- If the area enclosed by the two functions and is what is the value of ?
- Find the horizontal line that divides the region between and into two equal areas.