1. 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

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

3. The Fundamental Theorem of Calculus Part 2 & U-Sub
Chapter 5.3 & 5.5
February 6, 2007

4. 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

5. Fundamental Theorem of Calculus (Part 1)

6. 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!

7. Example

8. Example We have to find an antiderivative; evaluate at 3;
subtract the results.

9. Example

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

11. Example

12. Example

13. Example

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

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

16. Example

17. Example

18. Example
Alternate notation

19. Example

20. Example

21. Example
= –1

22. Example
= –1
= 1

23. Example

24. Example

25. Example

26. Evaluate:

27. Evaluate:
Improper = 0)

28. Evaluate:

29. Given:
Write a similar expression for

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

31. Evaluate:
Multiply out:
Use FTC 2 to Evaluate:

32. 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)

33. Chain Rule for Derivatives:
Chain Rule backwards for Integration:

34. Look for:

35. Back to Our Example
Let

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

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

38. Our Original Example of 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]

39. 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

40. In-Class Assignment
Integrate using two different methods:
1st by multiplying out and integrating
2nd by u-substitution
Do you get the same result? (Don’t just assume or claim you do; multiply out your results to show it!)
If you don’t get exactly the same answer, is it a problem? Why or why not?