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AP Calculus

AP Calculus. Area. Area of a Plane Region. Calculus was built around two problems Tangent line Area. Area. To approximate area, we use rectangles More rectangles means more accuracy. Area. Can over approximate with an upper sum Or under approximate with a lower sum. Area.

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AP Calculus

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  1. AP Calculus Area

  2. Area of a Plane Region • Calculus was built around two problems • Tangent line • Area

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

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

  5. Area • Variables include • Number of rectangles used • Endpoints used

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

  7. 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.

  8. 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 .

  9. Area Approximation

  10. 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

  11. Area Approximation

  12. Area Approximation • So the width of each rectangle is

  13. So the width of each rectangle is Area Approximation • The height of each rectangle is either or

  14. 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

  15. 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

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