## First Application of Definite Integral

Read this section to see how some applied problems can be reformulated as integration problems. Work through practice problems 1-4.

### Area between f and g

We have already used integrals to find the area between the graph of a function and the horizontal axis.
Integrals can also be used to find the area between two graphs.
If for all in [a,b], then we can approximate the area
between and by partitioning the interval [a,b] and forming a
Riemann sum (Fig. 2). The height of each rectangle is
so the area of the i^{th} rectangle is . This approximation
of the total area is

The limit of this Riemann sum, as the mesh of the partitions approaches 0, is the definite integral .

We will sometimes use an arrow to indicate "the limit of the Riemann sum as the mesh of the partitions approaches zero," and will write

**Example 1: **Find the area bounded between the graphs of and for (Fig. 3).

Solution: It is clear from the figure that the area between and g is square inches. Using the theorem,
area between and for is

, and area between f and g for is .

The two integrals also tell us that the total area between and is 2.5
square inches.

The single integral hich is not the **area** we want in this problem. The value
of the** integral is 1.5**, and the value of the** area is 2.5**.

**Practice 1: **Use integrals and the graphs of and to determine the area between the graphs of and for .

**Example 2: **Two objects start from the same location and travel along the same
path with velocities and meters per second (Fig. 4). ow far ahead is after 3 seconds? After 5 seconds?

Solution: Since , the "area" between the graphs of and represents the distance between the objects.

After 3 seconds, the distance apart

After 5 seconds, the distance apart =

If , we can use the simpler argument that the area of region is and the area of
region is , so the area of region C, the area between and , is .

If the same function is not always greater, then we need to be very careful and find the intervals where and the intervals where .

**Example 3:** Find the area of the shaded region in Fig. 5.

Solution: For so the **area **of is