Calculus
Worksheet Areas and Volumes
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 subintervals defined by the xinter
cepts 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.
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 Spring '08
 Grether
 Calculus, xaxis

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