Section 6.1 – Finding The Area
Between Two Curves
·
In order to find the area
between two curves, lines, etc, you must use integrals in order find the
specified area asked.
·
When using it in math terms,
you will take your equations and or lines and take whatever is on top minus
whatever is on bottom. For example, if you have the equations
and
, 
is the top equation
and 
is the bottom
equation when graphed.
·
Once you find this out, you
set up your integral (top – bottom). If there is only one equation given in the
problem, you do not do top equation – bottom equation.
·
Integral limits are in terms
of x and if your limits are not given in the problem, then you can set your
equations equal to each other if possible, or you can graph it.
For area, the integral equation
is:
à f(x) is the top equation and g(x) is the bottom equation. If there is
only one equation in the problem then you do the integral only for f(x).
After setting up the integral for
the problem, you will do the anti-derivative of what’s inside. Once you have
the anti-derivative, you will plug in your limits expressed (top limit first,
bottom limit second). After plugging in the limits and solving, the area
between the curves and lines are expressed.
Examples:
1) This example, there are two equations expressed, however
limits are not expressed. You can graph the equations to see which equations
are on top and which are on bottom.
- Here are your given 2 equations.
- Set equations equal to each
other to find limits.
- Take the top equation – the
bottom equation and take the anti-derivative. Then plug in your limits to find
the area.
2) In this problem, you are given two equations and lines to
set a distinct area. When you have two equations, you must remember to see
which equation is on top and bottom, you can do this by graphing the lines in
your calculator. Limits are expressed by the lines x = 0 and x = 1.
- These are the equations given. Limits are from 0 to 1.
- Then you do the
anti-derivative of the two equations and plug in the limits to find area.
3) In this problem, you are given two equations but neither is
considered top or bottom since the lines are both above each other at the same
height. In order to find the area, you need to split the integral.
- With this example, there is no top or bottom equation.
- Thus, it must split up
into different integrals.
Here are some sites that may offer additional help:
http://tutorial.math.lamar.edu/AllBrowsers/2413/AreaBetweenCurves.asp
http://archives.math.utk.edu/visual.calculus/5/area2curves.3/4.html