We have already learned how to find the area under a curve by using integrals. However, now we will be finding the aread between curves as indicated by the title of the post.
Let’s illustrate this idea:
So far we have done the area under a curve which looks like this:
And like this:
But now we are learning how to find the area between f(x) and g(x).
The equation for this process is
With f(x) being the curve on top, like the red curve in the pictures above.
This looks like…
Warning! Be careful when it is hard to distinguish which function is the top one!
Now let’s try to do it by hand.
Find the area of the region bound by the following
f(x) = -x^2 + 4x + 1
g(x) = x + 1
First set up the integral equation:
Next, do the simple math and combine like terms. It should be
Next take the integral
Now, evaluate on the interval [0,3]
= 9/2 or 4.5
Good job. Now let’s try it calculator style.
1. Graph both equations in Y one and Y two
2. Take the integral of the top ( 2nd Calc, 7, lower limit, upper limit)
Take the integral of the bottom, and subtract the bottom from the top.
Now you have your integral the long way on the calculator. But there is good news, there is an easier way!
1. In Y three put “Yone minus Ytwo” (if the equation in Yone is the curve on top)
2. Graph, there should be another curve. This curve represents the area of f(x) – g(x). Isn’t that what we are trying to find? Yes it is. soooo…
3. Take the integral (2nd calc 7), but! hit the down arrow until in the top left corner you see your equation for Yone minus Ytwo. Enter the lower limit and the upper limmit. The integral of the new curve is taken and whala you have your answer. This will be eaiser to use for those equations who don’t have rational integrals.
Homework p. 418 (1-19)O 31, 33, 53, 63
