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Exploring Integral Calculus: The Definite Integral and Fundamental Theorem

Delve into estimating with finite sums, definite integrals, antiderivatives, and the Fundamental Theorem of Calculus to master integral calculus fundamentals. Learn through examples, properties, and applications. Build the groundwork for advanced calculus concepts.

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Exploring Integral Calculus: The Definite Integral and Fundamental Theorem

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  1. Chapter 5 The Definite Integral

  2. 5.1 Estimating with Finite Sums

  3. Quick Review

  4. Quick Review Solutions

  5. 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.

  6. Example Finding Distance Traveled when Velocity Varies

  7. Example Finding Distance Traveled when Velocity Varies

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

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

  10. 5.2 Definite Integrals

  11. Quick Review

  12. Quick Review Solutions

  13. 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.

  14. Sigma Notation

  15. The Definite Integral as a Limit of Riemann Sums

  16. The Existence of Definite Integrals

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

  18. The Definite Integral

  19. Example Using the Notation

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

  21. Area

  22. The Integral of a Constant

  23. Example Using NINT

  24. 5.3 Definite Integrals and Antiderivatives

  25. Quick Review

  26. Quick Review Solutions

  27. 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.

  28. Rules for Definite Integrals

  29. Example Using the Rules for Definite Integrals

  30. Example Using the Rules for Definite Integrals

  31. Example Using the Rules for Definite Integrals

  32. Average (Mean) Value

  33. Example Applying the Definition

  34. The Mean Value Theorem for Definite Integrals

  35. The Mean Value Theorem for Definite Integrals

  36. The Derivative of an Integral

  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

  42. Quick Quiz Sections 5.1 - 5.3

  43. 5.4 Fundamental Theorem of Calculus

  44. Quick Review

  45. Quick Review Solutions

  46. What you’ll learn about • Fundamental Theorem, Part 1 • Graphing the Function • Fundamental Theorem, Part 2 • Area Connection • Analyzing Antiderivatives Graphically … and why The Fundamental Theorem of Calculus is a Triumph of Mathematical Discovery and the key to solving many problems.

  47. The Fundamental Theorem of Calculus

  48. The Fundamental Theorem of Calculus

  49. Example Applying the Fundamental Theorem

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