AP Calculus

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AP Calculus

• Area

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Area of a Plane Region

• Calculus was built around two problems
• Tangent line
• Area

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Area

• To approximate area, we use rectangles
• More rectangles means more accuracy

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Area

• Can over approximate with an upper sum
• Or under approximate with a lower sum

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Area

• Variables include
• Number of rectangles used
• Endpoints used

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Area

• Regardless of the number of rectangles or types
  of inputs used, the method is basically the same.
• Multiply width times height and add.

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Upper and Lower Sums

• An upper sum is defined as the area of
  circumscribed rectangles
• A lower sum is defined as the area of inscribed
  rectangles
• The actual area under a curve is always between
  these two sums or equal to one or both of them.

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Area Approximation

• We wish to approximate the area under a curve f
  from a to b.

• We begin by subdividing the interval a, b into
  n subintervals.

• Each subinterval is of width .

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Area Approximation
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Area Approximation

• We wish to approximate the area under a curve f
  from a to b.

• We begin by subdividing the interval a, b into
  n
• subintervals of width .

Minimum value of f in the ith subinterval
Maximum value of f in the ith subinterval
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Area Approximation
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Area Approximation

• So the width of each rectangle is

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Area Approximation

• So the width of each rectangle is

• The height of each rectangle is either

or
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Area Approximation

• So the width of each rectangle is

• The height of each rectangle is either

or

• So the upper and lower sums can be defined as

Lower sum Upper sum
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Area Approximation

• It is important to note that
• Neither approximation will give you the actual
  area
• Either approximation can be found to such a
  degree that it is accurate enough by taking the
  limit as n goes to infinity
• In other words