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. Fundamental Theorem of Calculus (Part 1) (Chain Rule)
Chapter 5.3 & 5.5
February 6, 2007
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

3. Fundamental Theorem of Calculus (Part 1)

4. 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!

5. Example

6. Example

7. Evaluate:

8. Evaluate:

9. Evaluate:

10. Given:
Write a similar expression for

11. Evaluate:
Multiply out:

12. Chain Rule for Derivatives:
What if instead?
Chain Rule for Derivatives:
Chain Rule backwards for Integration:
Look for:

13. Back to Our Example
Let

14. The same substitution holds for the higher power!
With

15. Our Original Example of a Definite Integral:
To make the substitution complete for a Definite Integral:
We make a change of bounds using:

16. 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

17. 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?