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 toa roximate the area under a curve from a to b. We begin by subdividing the interval [a, b] into n subintervals. Each subinterval is of width Ax.
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a Area Approximation b
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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, bl into n subintervals of width n Minimum value offin the ith subinterval f (Ml) = Maximum value offin the ith subinterval
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Area Approximation AMI): f(mı): a Ax b
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Area Approximation So the width of each rectangle is Ax = n i
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AMI): f(mı): a b
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Area Approximation So the width of each rectangle is Ax = The height of each rectangle is either f (m) i or f(Ml)
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AMI):
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Area Approximation So the width of each rectangle is Ax = The height of each rectangle is either f (m) i or f(Ml) So the upper and lower sums can be defined as Lower sum = s n i=l Upper sum = Ef(Mlbr
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Area Approximation It is important to note that s(n) (Area) S(n) 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 lim lim S(n)
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