Area Between Curves
In this section we are going to look at finding the area
between two curves. There are actually two cases that we are going to be
looking at.
In the first case we are want to determine the area between
 and
 on
the interval [a,b]. We are also going to assume that
 .
Take a look at the following sketch to get an idea of what we�re initially going
to look at.
In the
Area and Volume Formulas section of the Extras chapter we derived the
following formula for the area in this case.
The second case is almost identical to the first case.
Here we are going to determine the area between
 and
 on
the interval [c,d] with
 .
In this case the formula is,
Now
(1) and
(2) are perfectly serviceable formulas, however, it is sometimes easy to
forget that these always require the first function to be the larger of the two
functions. So, instead of these formulas we will instead use the following
�word� formulas to make sure that we remember that the formulas area always the
�larger� function minus the �smaller� function.
In the first case we will use,
|
(3) |
In the second case we will use,
|
(4) |
Using these formulas will always force us to think about
what is going on with each problem and to make sure that we�ve got the correct
order of functions when we go to use the formula.
Let�s work an example.
Example
Determine the area of the region enclosed by
 and
 .
Solution
First of all, just what do we mean by �area enclosed by�.
This means that the region we�re interested in must have one of the two curves
on every boundary of the region. So, here is a graph of the two functions with
the enclosed region shaded.
Note that we don�t take any part of the region to the right
of the intersection point of these two graphs. In this region there is no
boundary on the right side and so is not part of the enclosed area. Remember
that one of the given functions must be on the each boundary of the enclosed
region.
Also from this graph it�s clear that the upper function
will be dependent on the range of x�s that we use. Because of this you
should always sketch of a graph of the region. Without a sketch it�s often easy
to mistake which of the two functions is the larger. In this case most would
probably say that
 is
the upper function and they would be right for the vast majority of the x�s.
However, in this case it is the lower of the two functions.
The limits of integration for this will be the intersection
points of the two curves. In this case it�s pretty easy to see that they will
intersect at
 and
 so
these are the limits of integration.
So, the integral that we�ll need to compute to find the
area is,
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