Finding the Area Between Curves

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Finding the Area Between Curves

• Application of Integration

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Notes to BC students

• I hope everyone had great holidays, I did,
  including experiencing a blizzard, but now Im
  sick

• Since we missed the time before the holidays,
  some Unit 6 topic(s) will be moved to Quarter III.

• This applies to both morning and afternoon
  classes.

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• The problem is to find the area between two
  curves, so we start with a couple of friendly
  calculus curves.

The first is , or .
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• And the second is

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• A closer look

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• We are interested in finding the area of the
  purple region.

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• Let h be the distance between the two curves.

h
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• Notice how h changes as we move from left to
  right.

h
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Since h is the distance from the upper to lower
curve. This is simply the difference of the two
y-coordinates.
This means that
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• We can find the total area between the curves by
  integrating h between the points of intersection.

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• Note that the two curves intersect at the origin
  and at (1,1).

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The area between the curves is
The 0 and 1 are the starting and ending values of
x.
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Further,
The area is
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We can evaluate the integral using the
Fundamental Theorem of the Calculus.
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As a second example, find the area between
First, we need to graph the functions and see the
defined area.
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Zooming in
Notice that the upper intersection is not made of
simple values.
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Later, we will find the intersection. First, we
define h.
Notice that h is the difference between the two
x-coordinates.
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Notice this distance uses coordinates from the
right function minus coordinates from the left
function.
To have distance be a positive number one must
always subtract a smaller from a larger one.
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As with the first example we integrate h from
beginning to end. We see that the origin is one
point of intersection.
We need to find the other point of intersection.
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Finally, the area is
This is a good time to use your calculator!
Note that in this example the limits of
integration are y-values, and the integrand is a
function of y.
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There are several points that should be made

• Graph the functions.

• Decide whether you will work in vertical or
  horizontal distances. Use the one that it
  easiest for the problem. n.b. This is not always
  x!

• Distance is always positive, remember to subtract
  the smaller value from the larger one, whether
  using x or y.