Area &Distances
Section 5.1
Consider the function fx1
xover the interval from 1 to 5. What is the area under this
section of the curve and above the x-axis?
0 1 2 3 4 5
0.0
0.5
1.0
One way we can estimate the area under the curve break up the interval into "strips" (what
we call a partition) and then try and approximate the area of the resulting strips. We could
then, for instance, use rectangles to approximate each strip and then add all the areas up to
obtain an approximation of the total area.
Let’s do this using strips of equal widths of 1 unit each. This will give us 4 strips. How do
we draw each rectangle? Let’s use the value of fat the right endpoints of each interval:
f21
2f31
3f41
4f51
5
The rectangles would look like this:
0 1 2 3 4 5
0.0
0.5
1.0
x
y
The area of these rectangles will be designated R4(Rfor right endpoints & 4 for the
number of strips):
R411
211
311
411
577
60 ≈1.283333
This area represents an UNDER estimate for the true area.
We could also obtain an OVER estimate by using the value of fat the left endpoints.
f11
11f21
2f31
3f41
4
The rectangles would look like:
0 1 2 3 4 5
0.0
0.5
1.0
x
y
