AREA UNDER A CURVE
It’s time to learn one of the most important uses of the integral. We’ve already discussed how integration
can be used to “antidifferentiate” a function; now you’ll see that you can also use it to find the area under
a curve. First, here’s a little background about how to find the area without using integration.
Suppose you have to find the area under the curve y = x^2 + 2 from x = 1 to x = 3. The graph of the curve
looks like the following:
Don’t panic yet. Nothing you’ve learned in geometry thus far has taught you how to find the area of
something like this. You have learned how to find the area of a rectangle, though, and we’re going to use
rectangles to approximate the area between the curve and the x-axis.
Let’s divide the region into two rectangles, one from x = 1 to x = 2 and one from x = 2 to x = 3, where the
top of each rectangle comes just under the curve. It looks like the following: