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Understanding Definite Integrals and Area of Bounded Regions in Mathematics

This session focuses on the concept of definite integrals as limits of sums and explores the areas of bounded regions. Students will engage with class exercises designed to deepen their understanding through practical examples. Key topics include the definition of definite integrals, the areas delineated by curves, and the application of these integrals to solve geometric problems involving continuous functions. The session includes step-by-step examples that illustrate these concepts, enhancing comprehension and analytical skills in calculus.

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Understanding Definite Integrals and Area of Bounded Regions in Mathematics

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Presentation Transcript

  1. Mathematics

  2. Session Definite Integrals - 3

  3. Session Objectives • Definite Integral as the Limit of a Sum • Areas of Bounded Regions • Class Exercise

  4. Definite Integral as the Limit of a Sum OR

  5. Example - 1

  6. Solution Cont.

  7. Example - 2

  8. Solution Cont.

  9. Example - 3

  10. Solution Cont.

  11. Example - 4

  12. Solution Cont.

  13. Example - 5

  14. Solution Cont.

  15. Areas of Bounded Regions 1. Let f(x) be a continuous function defined on the interval [a, b]. Then, the area bounded by the curve y = f(x), x-axis and the ordinates x = a, x = b is • The area bounded by the curve x = f(y), y-axis and the abscissae y = c, y = d is

  16. Areas of Bounded Regions Cont.

  17. Example - 6

  18. Solution Cont.

  19. Example - 7

  20. Solution Cont.

  21. Solution Cont.

  22. Example - 8 Solution: The given curves are (i) and (ii) intersect at (1, 0) and (0, 1).

  23. Solution Cont.

  24. Example - 9

  25. Solution Cont.

  26. Example - 10 Solution: The given curves are

  27. Solution Cont.

  28. Solution Cont.

  29. Solution Cont.

  30. Example - 11

  31. y x2 + y2 = 1 x x' (1, 0) O (x-1)2 + y2 = 1 y' Solution Cont.

  32. Solution Cont.

  33. Solution Cont.

  34. Thank you

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