1. SYMMETRY Even and Odd Suppose f is continuous on [-a, a] and even
Suppose f is continuous on [-a, a] and odd

2. Even and Odd
Term-102

3. Even and Odd
Term-091

4. Even and Odd
Term-103

5. SYMMETRY Even and Odd Suppose f is continuous on [-a, a] and even
Suppose f is continuous on [-a, a] and odd
Example

6. Area between two curves
AREAS BETWEEN CURVES
Area between two curves

7. Example: Note: AREAS BETWEEN CURVES Area between two curves
Find the area of the region bounded by the curves
and
Note:
Both right-hand and left-hand boundary are lines

8. Area between two curves
AREAS BETWEEN CURVES
Area between two curves
Note:
Both right-hand and left-hand boundary reduce to a point

9. 1) Find the intersection points
AREAS BETWEEN CURVES
Note:
The value of a and b (not given)
Steps:
1) Find the intersection points
2) Write the area as an integral

10. Area between two curves
AREAS BETWEEN CURVES
Area between two curves

11. AREAS BETWEEN CURVES
T-092

12. AREAS BETWEEN CURVES
T-102

13. AREAS BETWEEN CURVES

14. Area between two curves
AREAS BETWEEN CURVES
Area between two curves
Note:
Note:
right-hand boundary = ??
left-hand boundary = ??
right-hand boundary = ??
left-hand boundary = ??
Note:
Note:
Top boundary = ??
Bottom boundary = ??
Top boundary = ??
Bottom boundary = ??

15. Area between two curves
AREAS BETWEEN CURVES
Area between two curves
Some regions are best treated by regarding x as a function of y.

16. AREAS BETWEEN CURVES
Some regions are best treated by regarding x as a function of y.

17. 1) Rewrite the function as x in terms of y
AREAS BETWEEN CURVES
Steps:
1) Rewrite the function as x in terms of y
2) Find the intersection points

18. AREAS BETWEEN CURVES
We could have found the area in by integrating with respect to x instead of y, but the calculation is much more involved.

19. AREAS BETWEEN CURVES
T-092

20. AREAS BETWEEN CURVES