Area of a Region

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In the previous pages you reviewed how to find the area under a graph. This page will now refresh your memeory with finding the area of a region between two curves. This skill is one that should be learned before taking the AP test.

Area of a Region Between Two Curves
If f and g are continuous on [a, b] and g(x) <  f(x) for all x in [a, b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is
         _b
A = §  [f(x) - g(x)] dx.
         ^a

Because many AP problems will have the curves intersecting, the example below will be a problem dealing with a region lying between two interseting graphs. Note that when the graphs do not intersect, the values of a and b are given explicitly.

Example: A Region Lying Between Two Intersecting Graphs

Situation: You must find the area of the region bounded by the graphs of f(x) = 2 - x² and g(x) = x.


Solution: Notice in the image of the two graphs of f and g. Both have two points of intersection. To find the x-coordinates of these points, set f(x) and g(x) equal to each other and solve for x.

Set f(x) equal to g(x)
2 - x² = x

Write them in standard form or, in other words, get everything on one side and set it to 0.
-x² - x + 2 = 0

Factor
-(x + 2)(x - 1) = 0

Solve for x
x = -2 or 1

As a result, a = -2 and b = 1. Because g(x) < f(x) on the interval [-2, 1], the representative rectangle has an area of A = [f(x) - g(x)]x = [(2 - x²) - x]x.
The area of the region is
         _{1                                                            1}
A = §  [(2 - x²) - x] dx = [-x³/3 - x²/2 + 2x]   = 9/2.
        ^{-2                                                           -2}