1. Sec 5.1: Areas and Distances
Example:
Example:
Example:
Formula:

2. Example: Example: Sigma Notation: Sec 5.1: Areas and Distances
This tells us to
end with i=100
This tells us
to add.
This tells us to
start with i=1

3. Example Formula Sec 5.1: Areas and Distances This tells us to
end with i=100
This tells us
to add.
This tells us to
start with i=1
Example
Formula

4. Sec 5.1: Areas and Distances

5. Sec 5.1: Areas and Distances
TERM-111
Exam-1

6. Sec 5.1: Areas and Distances
The Area Problem
Divide it into triangles
it isn’t so easy to find the area of a region with curved sides

7. Sec 5.1: Areas and Distances
The Area Problem
Find the area of the region that lies under the curve y=f(x) from a to b.
Curve y=f(x), vertical lines x=a and x=b and the -axis.

8. Example: Sec 5.1: Areas and Distances
Use rectangles to estimate the area under the parabola
from 0 to 1

9. Example: Sec 5.1: Areas and Distances height is the right endpoints
Use rectangles to estimate the area under the parabola
from 0 to 1
height is the right endpoints

10. Example: Sec 5.1: Areas and Distances height is the left endpoints
Use rectangles to estimate the area under the parabola
from 0 to 1
height is the left endpoints

11. Example: Sec 5.1: Areas and Distances
Use rectangles to estimate the area under the parabola
from 0 to 1

12. Example: Sec 5.1: Areas and Distances
Use rectangles to estimate the area under the parabola
from 0 to 1
We can repeat this procedure with a larger number of strips
<S <

13. Example: Sec 5.1: Areas and Distances
Use rectangles to estimate the area under the parabola
from 0 to 1
We could obtain better estimates by increasing the number of strips

14. Sec 5.1: Areas and Distances
Example:

15. Sec 5.1: Areas and Distances
Let’s apply the idea to the more general region S
We start by subdividing the interval [a,b] into n subintervals
The width of the interval [a,b] is b-a
the width of each subinterval is
The subintervals are

16. Sec 5.1: Areas and Distances

17. Sec 5.1: Areas and Distances
Example:

18. Sec 5.1: Areas and Distances
Example:

19. Sec 5.1: Areas and Distances

20. Note: Area under a curve = limit of summation
Sec 5.1: Areas and Distances
Note: Area under a curve = limit of summation
Note:

21. Sec 5.1: Areas and DistancesHW HW

22. Sec 5.1: Areas and Distances
EXAM-1
TERM-102

23. Sec 5.1: Areas and Distances
THE DISTANCE PROBLEM
Find the distance traveled by an object during 10 seconds if the velocity of the object is 35 ft/s
distance = velocity x time
Here the velocity is constant
But if the velocity varies, it’s not so easy to find the distance traveled.

24. Sec 5.1: Areas and Distances
THE DISTANCE PROBLEM