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  1. 5.1 Estimating with Finite Sums

  2. Quick Review

  3. Quick Review Solutions

  4. What you’ll learn about • Distance Traveled • Rectangular Approximation Method (RAM) • Volume of a Sphere • Cardiac Output … and why Learning about estimating with finite sums sets the foundation for understanding integral calculus.

  5. Example Finding Distance Traveled when Velocity Varies

  6. Example Finding Distance Traveled when Velocity Varies

  7. LRAM, MRAM, and RRAM approximations to the area under the graph of y=x2 from x=0 to x=3

  8. Example Estimating Area Under the Graph of a Nonnegative Function

  9. 5.2 Definite Integrals

  10. Quick Review

  11. Quick Review Solutions

  12. What you’ll learn about • Riemann Sums • The Definite Integral • Computing Definite Integrals on a Calculator • Integrability … and why The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus.

  13. Sigma Notation

  14. The Definite Integral as a Limit of Riemann Sums

  15. The Existence of Definite Integrals

  16. The Definite Integral of a Continuous Function on [a,b]

  17. The Definite Integral

  18. Example Using the Notation

  19. Area Under a Curve (as a Definite Integral)

  20. Area

  21. The Integral of a Constant

  22. Example Using NINT

  23. 5.3 Definite Integrals and Antiderivatives

  24. Quick Review

  25. Quick Review Solutions

  26. What you’ll learn about • Properties of Definite Integrals • Average Value of a Function • Mean Value Theorem for Definite Integrals • Connecting Differential and Integral Calculus … and why Working with the properties of definite integrals helps us to understand better the definite integral. Connecting derivatives and definite integrals sets the stage for the Fundamental Theorem of Calculus.

  27. Rules for Definite Integrals

  28. Example Using the Rules for Definite Integrals

  29. Example Using the Rules for Definite Integrals

  30. Example Using the Rules for Definite Integrals

  31. Average (Mean) Value

  32. Example Applying the Definition

  33. The Mean Value Theorem for Definite Integrals

  34. The Mean Value Theorem for Definite Integrals

  35. The Derivative of an Integral

  36. Quick Quiz Sections 5.1 - 5.3

  37. Quick Quiz Sections 5.1 - 5.3

  38. Quick Quiz Sections 5.1 - 5.3

  39. Quick Quiz Sections 5.1 - 5.3

  40. Quick Quiz Sections 5.1 - 5.3

  41. Quick Quiz Sections 5.1 - 5.3