rea and Volume Formulas
In this section we will derive the formulas used to get the area between two curves
and the volume of a solid of revolution.
Area Between Two Curves
We will start with the formula for determining the area between
and
on the interval [
a,b
]. We will also assume
that
on [
a,b
].
We will now proceed much as we did when we looked that the
Area Problem
in the
Integrals Chapter. We will first divide up the interval into
n
equal subintervals each
with length,
Next, pick a point in each subinterval,
, and we can then use rectangles on each
interval as follows.
The height of each of these rectangles is given by,
and the area of each rectangle is then,
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So, the area between the two curves is then approximated by,
The exact area is,
Now, recalling the
definition of the definite integral
this is nothing more than,
The formula above will work provided the two functions are in the form
and
. However, not all functions are in
that form. Sometimes we will be forced to work with functions in the form
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- Fall '08
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- Formulas
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