1. Review
Calculus

2. Example 1: A) B) C)

3. Example 1 (continue) D)

4. Example 1 (continue) E)

5. Example 2 A)

6. Example 2 (continue) B)

7. Example 2 (continue) C)

8. Example 2 (continue) D)

9. Example 2 (continue) E)

10. Evaluate the definite integral and check the result by differentiation.
Rewrite as

11. Evaluate the definite integral and check the result by differentiation.
Rewrite:
Integrate:
Simplify:

12. The variable of integration must match the variable in the expression.
Example
The variable of integration must match the variable in the expression.
Don’t forget to substitute the value for u back into the problem!

13. Substitution rule examples
We computed du by straightforward differentiation of the expression for u.
The substitution u = 2x was suggested by the function to be integrated. The main problem in integrating by substitution is to find the right substitution which simplifies the integral so that it can be computed by the table of basic integrals.

14. Example about choosing the substitution
Solution
This rewriting allows us to finish the computation using basic formulae.
Next substitute back to the original variable.

15. The substitution rule for definite integrals
Example 1 e
The area of the yellow domain is ½.

16. Don’t forget to use the new limits.
Example
Don’t forget to use the new limits.

17. Examples – INTEGRATING EXPONENTIAL FUNCTIONS

18. The Integral
The integral can be written in three different ways, all of which have the same answer.

19. Examples 1. ∫ 1/x dx = ln |x| + C, x ≠ 0. ∫ x - 1 dx = 2. ∫ 4/x dx =

20. Example 1 – Solution

21. Example: Find the area of the region that is bounded by the graphs of
First, look at the graph
of these two functions.
Determine where they
intersect.
(endpoints not given)

22. Example (continued):
Second, find the points of intersection by setting f (x) = g (x) and solving.

23. ò Example (concluded):
Lastly, compute the integral. Note that on [0, 2], f (x) is the upper graph. ( 2 x + 1 ) - é ë ù û ò d = 3 ê ú æ è ç ö ø ÷ 4 8

24. Model Problem
Find the area of the region bounded by the graphs of y = x2 + 2, y = -x, x = 0, and x = 1.
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25. Model Problem
Find the area of the region bounded by the graph of f(x) = 2 – x2 and g(x) = x.
a and b ?
points of intersection

26. Aim: How do we find the area of a region between two curves?
Do Now:
Find the area of the region between the graphs of f(x) = 1 – x2 and g(x) = 1 – x.

27. Model Problems
Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x.
points of intersection

28. Model Problems
Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x.