Area Under the Curve

1
Area Under the Curve

• We want to approximate the area between a curve
  (yx21) and the x-axis from x0 to x7
• We will use rectangles to do this.
• One way will be to choose rectangles whose
  heights are taken from the x-coordinate of the
  right side of the rectangle (Right Sum)
• We will let n be the number of rectangles we use
  to approximate the area

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Right Sum n5
3
Right Sum n10
4
Right Sum n20
5
Right Sum n50
6
Right Sum n100
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What do you notice?

• What will happen as you add more rectangles for a
  Right Sum?
• Will this happen for any function? Why or why
  not?
• How many rectangles do we need to get the actual
  area?
• Can you think of another way to approximate the
  area?

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Another way

• What if we were to use rectangles whose heights
  were formed from the x-coordinate of the Left
  side of the interval?
• We will call these Left Sums
• As we go through the Left Sums, what do you
  notice about the areas?

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Left Sum n5
10
Left sum n10
11
Left Sum n20
12
Left Sum n50
13
Left Sum n100
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Lets look at the areas again
Right Sums Left Sums
n5 157.92 89.32
n10 139.055 104.755
n20 130.0513 112.9013
n50 124.7862 117.9262
n100 123.054 119.624
n1000 121.5049 121.1619
n5000 121.3676 121.299
n30000 121.3391 121.3276
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Another Example

• While it is easiest for computational reasons to
  look at Left Sums and Right Sums, theoretically
  it is necessary to look at Upper Sums (where each
  rectangle circumscribes the function) and Lower
  Sums (where each rectangle is inscribed in the
  function). Recall that in the example seen so
  far, the Right Sums were also Upper Sums and the
  Left Sums were Lower Sums. Under what
  circumstances would this not be true? Is it
  possible for some Upper Sums to be left sums and
  others to be right sums?

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Here is an example of an Upper Sum. Notice that
the rectangles are all formed by choosing the
height from the highest point on the graph that
the rectangle hits.
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Here is a Lower Sum with 8 rectangles. What do
you think will happen if a rectangle starts at
-½ and ends at ½? What would the height be for
a Lower Sum?
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Finding Exact AreasRiemann Sums

• It turns out, as you could see from the table,
  that if you use enough rectangles, the Left Sum
  will be very close to the Right Sum. If you use
  an infinite amount of rectangles, the Left Sum
  and Right Sum will be equal, and they will equal
  the exact area.
• Theoretically, there are a great many ways to
  design rectangles that will lead to the area.
  Any of these will be a type of Riemann Sum. So
  far we have seen Left, Right, Upper, and Lower
  Sums these are all types of Riemann Sums. There
  are lots of others.
• Since any Riemann Sum will eventually lead to the
  exact area, we will use the Right Sum. The Right
  Sum is computationally easiest to use.

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Finding Exact AreasRiemann Sums

• Back to
• If we use 8 rectangles, how wide is each one?
• What about if we used 12 rectangles?
• How about n rectangles?
• In general, if we start at xa and stop at xb,
  how wide would each rectangle be?

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Exact Area

• In general, the width of each rectangle will be
• We call the width of each rectangle
• What would the height of each rectangle be? Lets
  look at the graph again, and then see if we can
  generalize

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As you can see, the heights of each rectangle are
found by getting the y-value at the right side of
each rectangle. If xi represents the x-coordinate
of the right side of the ith rectangle, then
the height of the ith rectangle is f(xi ). For
our example, each xi is found by adding to
the previous right side. Since we start at x0,
the first right side is the next one is
. Without doing the addition, what would be
the coordinate of the right side of the 5th
rectangle? Do you see that you merely multiply 5
times ? How would this generalize?
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• If the area we are interested starts at xa, and
  the width is then the x-coordinate
  of the ith rectangle xi will be
• Example If I was finding the area under a curve
  on the interval 3, 7 and I was using 100
  rectangles, the x-coordinate of the 70th
  rectangle would be
• Find x25 for an area on the interval 2, 8 if
  we use 120 rectangles
• Did you get 3.25?

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Area of the i th rectangle

• Since area of a rectangle is height times width,
  and the height is just the value of the function
  at xi , we get
• Given on the
  interval 5, 7 with 20 rectangles, find the area
  of the 14th rectangle.
• Did you get

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Putting it all together

• Now we want to put it all together. We want to
  add up all n rectangles to give an approximation
  for the area. This is the formula we use
• Lets go back to our first example
  on 0, 7 and lets use 100 rectangles. We
  get
• which is what we had before on our table.

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The exact area

• To find the exact area, all we need to do is look
  at an infinite number of rectangles. Believe it
  or not, this is actually easier than what we just
  did. The formula becomes
• The previous example becomes
• Remember that the sum of a constant is the
  constant times n.

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Another example

• Find the area under on
  the interval 2, 5
• We have ,
  and
• So the area is

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A few last comments

• Dont Panic an easier way is coming soon
• There are many, many applications of what we just
  didalso coming soon
• What do you think would change if we tried to
  find the exact area using a Left Sum? Why is
  using a Left Sum more complicated?
• Another, very common method used to approximate
  the area under a curve is called a Midpoint Sum.
  Without any other information, what do you think
  that might be?
• To approximate the area under a curve, we usually
  just use a few rectangles, and which method we
  use depends on what the graph looks like. To get
  the exact area we use Right Sums, but we are
  limited to only finding the area under polynomial
  of degree 3 or less (Why?)

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A final comment or three

• If you are approximating the area under a curve
  and you are using only a few rectangles it is
  MUCH easier to find the heights and areas by hand
  (or using a table in your calculator) than to use
  formulas.
• The width of each rectangle will not always be
  the samethink about how you approach that
  situation.
• On an AP test you will need to do Left, Right,
  Upper, Lower, and/or Midpoint Sums for small
  numbers of rectangles.