## Introduction to Integration

Read this section to learn about area. Work through practice problems 1-9.

### Area

The basic shape we will use is the rectangle; the area of a rectangle is If the units for each side of the rectangle are "meters," then the area will have the units . The only other area
formulas needed for this section are for triangles, , and for circles, . Three other familiar properties of area are assumed and will be used:

**Addition Property: **The total area of a region is the sum of the areas of the non–overlapping pieces which comprise the region. (Fig. 1)

**Inclusion Property:** If region B is on or inside region A, then the area of region B is less than or equal to the area of region A. (Fig. 2)

**Location–Independence Property:** The area of a region does not depend on its location. (Fig. 3)

**Example 1:** Determine the area of the region in Fig. 4a

Solution: The region can easily be broken into two rectangles, Fig. 4b, with areas 35 square inches and 3 square inches respectively, so the area of the original region is 38 square inches.

**Practice 1: **Determine the area of the region in Fig. 5 by cutting it in two ways: (a) into a rectangle and triangle and (b) into two triangles.

We can use the three properties of area to get information about areas that are difficult to calculate exactly. Let be the region bounded by the graph of , the x–axis, and vertical lines at and . Since the two rectangles in Fig. 6 are inside the region A and do not overlap each, the area of the rectangles, , is less than the area of region A.

**Practice 2: **Build two rectangles, each with base 1 unit, outside the shaded region in Fig. 6 and use their areas to make a VALID statement about the area of region A.

**Practice 3: **What can be said about the area of region A in Fig. 6 if we use both inside and outside rectangles with base 1/2 unit?

**Example 2: **In Fig. 7, there are 32 dark squares, 1 centimeter on a side, and 31 lighter squares of the same size. We can be sure that the area of the leaf is smaller than what number?

Solution: The area of the leaf is smaller than .

**P****ractice 4: **We can be sure that the area of the leaf is at least how large?

Functions can be defined in terms of areas. For the constant function , define to be the area of the rectangular region bounded by the graph of , the t–axis, and the vertical lines at and (Fig. 8a). is the area of the shaded region in Fig. 8b, and . Similarly, and . In general, for any . The graph of is shown in Fig. 8c, and for every value of .

**Practice 5:** For , define to be the area of the region bounded by the graph of , the t–axis, and vertical lines at . Fill in the table in Fig. 9 with the values of . How are the graphs of related?

Sometimes it is useful to move regions around. The area of a parallelogram is obvious if we move the triangular region from one side of the parallelogram to fill the region on the other side and ending up with a rectangle (Fig. 10). At first glance, it is difficult to estimate the total area of the shaded regions in Fig. 11a . However, if we slide all of them into a single column (Fig. 11b), then it is easy to determine that the shaded area is less than the area of the enclosing .

**Practice 6: **The total area of the shaded regions in Fig. 12 is less than what number?