Area between two Curves

Figure 4.5.1

A region in the plane can often be represented as the region between two curves. For example, the unit circle is the region between the curves

shown in Figure 4.5.1. Consider two continuous functions f and g on [a, b] such that f (x) ≤ g(x) for all x in [a, b]. The region R, bounded by the curves

y = F(x), y = g(x), x = a, x = b,

is called the region between f (x) and g(x) from a to b. If both curves are above the x-axis as in Figure 4.5.2, the area of the region R can be found by subtracting the area below f from the area below g:

It is usually easier to work with a single integral and write

Figure 4.5.2

In the general case shown in Figure 4.5.3, we may move the region R above the x-axis by adding a constant c to both f(x) and g(x) without changing the area, and the same formula holds:

Figure 4.5.3

To sum up, we define the area between two curves as follows.

Example 1: Area Between Two Curves

Example 2: Area of a Region Bounded by Two Curves

Example 3: Area Below A Curve And A Line