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The FTC Part 2, Total Change/Area & U-Sub

The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1. Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow? A. log(7)/3 B. (1/3)log(13/7) C. 3 log (13/7) D. 3log(7) E. Never.

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The FTC Part 2, Total Change/Area & U-Sub

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  1. The FTC Part 2, Total Change/Area & U-Sub

  2. Question from Test 1 • Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow? • A. log(7)/3 • B. (1/3)log(13/7) • C. 3 log (13/7) • D. 3log(7) • E. Never

  3. Question from Test 1 • Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow? • A. log(7)/3

  4. The Total Change Theorem The integral of a rate of change is the total change from a to b. (displacement) (still from last weeks notes)

  5. The Total Change Theorem • Ex: Given • Find the displacement and total distance traveled from time 1 to time 6. • Displacement: (negative area takes away from positive) • Total Distance: (all area counted positive)

  6. Total Area • Find the area of the region bounded by the x-axis, y-axis and y = 2 – 2x. • First find the bounds by setting 2 – 2x = 0 and by substitution 0 in for x

  7. Total Area • Ex. Find the area of the region bounded by the y-axis and the curve

  8. Fundamental Theorem of Calculus (Part 1)(Chain Rule) • If f is continuous on [a, b], then the function defined by • is continuous on [a, b] and differentiable on (a, b) and

  9. Fundamental Theorem of Calculus (Part 1)

  10. Fundamental Theorem of Calculus (Part 2) • If f is continuous on [a, b], then : • Where F is any antiderivative of f. ( ) • Helps us to more easily evaluate Definite Integrals in the same way we calculate the Indefinite!

  11. Example

  12. Example • We have to • find an antiderivative; • evaluate at 3; • evaluate at 2; • subtract the results.

  13. Example

  14. Example This notation means: evaluate the function at 3 and 2, and subtract the results.

  15. Example

  16. Example

  17. Example

  18. Example Don’t need to include “+ C” in our antiderivative, because any antiderivative will work.

  19. Example the “C’s” will cancel each other out.

  20. Example

  21. Example

  22. Example Alternate notation

  23. Example

  24. Example

  25. Example = –1

  26. Example = –1 = 1

  27. Example

  28. Example

  29. Example

  30. Given: Write a similar expression for the continuous function:

  31. Fundamental Theorem of Calculus (Part 2) • If f is continuous on [a, b], then : Where F is any antiderivative of f. ( )

  32. Evaluate: • Multiply out: Use FTC 2 to Evaluate:

  33. What if instead? • It would be tedious to use the same multiplication strategy! • There is a better way! • We’ll use the chain rule (backwards)

  34. Chain Rule for Derivatives: • Chain Rule backwards for Integration:

  35. Look for:

  36. Back to Our Example • Let

  37. Our Example as anIndefinite Integral • With • AND Without worrying about the bounds for now: • Back to x (Indefinite):

  38. The same substitution holds for the higher power! • With • Back to x (Indefinite):

  39. Our Original Exampleof a Definite Integral: • To make the substitution complete for a Definite Integral: • We make a change of bounds using: • When x = -1, u = 2(-1)+1 = -1 • When x = 2, u = 2(2) + 1 = 5 • The x-interval [-1,2] is transformed to the u-interval [-1, 5]

  40. Substitution Rule for Indefinite Integrals • If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Substitution Rule for Definite Integrals • If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then

  41. Using the Chain Rule, we know that: Evaluate:

  42. Using the Chain Rule, we know that: Evaluate: Looks almost like cos(x2) 2x, which is the derivative of sin(x2).

  43. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

  44. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

  45. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

  46. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match: We put in a 2 so the pattern will match.

  47. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match: We put in a 2 so the pattern will match. So we must also put in a 1/2 to keep the problem the same.

  48. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

  49. Using the Chain Rule, we know that: Evaluate: We will rearrange the integral to get an exact match:

  50. Check Answer:

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