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Chapter 5 Integral

Chapter 5 Integral. Estimating with Finite Sums. Approach . Approach (2). Both approach are called Upper sum because they are obtained by taking the height of each rectangle as the maximum (uppermost) value of ƒ(x) for x a point in the base interval of the rectangle.

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Chapter 5 Integral

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  1. Chapter 5Integral

  2. Estimating with Finite Sums

  3. Approach

  4. Approach (2)

  5. Both approach are called Upper sum because they are obtained by taking the height of each rectangle as the maximum (uppermost) value of ƒ(x) for x a point in the base interval of the rectangle. • Now, we will be using what so called lower sum

  6. Therefore

  7. Midpoint approach

  8. Conclusions:

  9. Distance travelled • Suppose we know the velocity function y(t) of a car moving down a highway, without changing direction, and want to know how far it traveled between times t=a and t=b • If we already known an antiderivative F(t) of v(t) we can find the car’s position function s(t) by setting s(t)=F(t)+C. • The travelled distance is s(b)-s(a) • How to calculate in case we have no formula s(t)? • We need an approach in calculating s(t)

  10. approach • Subdivide the interval [a, b] into short time intervals on each of which the velocity is considered to be fairly constant. • distance = velocity x time • Total distance

  11. Average Value of a Nonnegative Function

  12. Sigma Notation and Limits of Finite Sums

  13. Limits of Finite Sums Solution: We start by subdividing [0, 1] into n equal width subintervals The lower sum of rectangular is :

  14. Riemann Sums

  15. Riemann Sums(2) the width of the kth subinterval is

  16. Riemann Sums(3)

  17. Riemann Sums (4) Among three figures, which one gives us the most accurate calculation?

  18. Riemann Sums (5) • In previous calculation, we can improve accuracy by increasing number of interval (n). • However, in Reimann sum, we can go to more accurate calculation by making |P| goes to zero • We define the norm of a partition P, written |P| to be the largest of all the subinterval widths. If |P| is a small number, then all of the subintervals in the partition P have a small width.

  19. The Definite Integral

  20. Notation and existence of definite Integrals

  21. Properties of Definite Integrals

  22. Average Value of a Continuous Function Revisited

  23. The Fundamental Theorem of Calculus

  24. The Fundamental Theorem of Calculus (2) But remember this

  25. Indefinite Integrals and the Substitution Rule • Symbol 

  26. Substitution: Running the Chain Rule Backwards

  27. Definite Integrals of Symmetric Functions

  28. Areas Between Curves

  29. But, be careful with this circumstances

  30. Integration with Respect to y

  31. Example : previous problem, but integration respects to y

  32. Combining Integrals with Formulas from Geometry

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