AP Calculus AB
Worksheet Areas, Volumes, and Arc Lengths
Areas
To find the area between the graph of f(x) and the xaxis from x = a to x = b we
first determine if the function crosses the xaxis on the interval.
If it does not cross the
xaxis and f(x) > 0 on the interval then the area is given by
Area =
Ÿ
a
b
f
H
x
L
„
x
If
the graph of f does not cross the xaxis and f(x) < 0 on the interval then the
area is given by
Area = 
Ÿ
a
b
f
H
x
L
„
x
If the graph of f crosses the xaxis we find the area by integrating on the subinter
vals defined by the xintercepts and adding the opposite of the integrals for any region
that is under the xaxis.
Note:
When using your calculator you can take a shortcut
when finding the area between the graph of a function and the xaxis by integrating the
absolute value of f(x) on the interval in question.
This results in the area between the
curve and the xaxis because the graph of the absolute value of f(x) will lie entirely
above the xaxis.
To find the area between two curves we find the points of intersection.
If they
intersect in only two points then one of the functions will dominate the other on that
interval.
The height of a typical rectangular element will then be the top function minus
the bottom function.
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 Spring '08
 Grether
 Arc Length, xaxis

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