Page 78 - Calc for the Clueless

Chapter 5
Integral Applications

Areas

We would like to explore some applications of the integral. The first is the area between two curves. Suppose
we have two functions f(x) and g(x), where f(x) is always greater than or equal to g(x). Its picture might look
like this:

We are going to add these rectangles up. One rectangle is represented by the height times the base = [f(x k) -
g(x k)]∆x k. If we add them up and take the limits properly, we get

Example 1—

2
Find the area between y = x + 1 and y = 3 - x .
We first draw the curves to see which is the top curve and which is the bottom curve. We next find the limits of
integration, the left- and rightmost x values, by setting the curves equal to each other to find the points where
they meet.

2
2
x + l = 3 - x , x + x - 2 = 0, (x + 2)( x - 1) = 0. So x = -2 and 1. We then set up the integral top-curve-minus-
bottom-curve dx from -2 to 1.