1. Sec 5.1: Areas and Distances

2. Sec 5.1: Areas and Distances

3. 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

4. Sec 5.1: Areas and Distances
Find the area of the region that lies under the curve
from to
Use two rectangles and left endpoints

5. Sec 5.1: Areas and Distances

6. Sec 5.1: Areas and Distances
Two rectangles with left endpoint
4 rectangles with left endpoint
Example:
Use rectangles to estimate the area under the parabola
from 0 to 1
4 rectangles with right endpoint
4 rectangles with midpoint

7. Sec 5.1: Areas and Distances
4 rectangles with right endpoint
4 rectangles with midpoint
4 rectangles with left endpoint
Estimating area with 4 rectangles using left end points
Estimating area with 4 rectangles using right end points
Estimating area with 4 rectangles using midpoints
Notations:

8. Sec 5.1: Areas and Distances

9. Sec 5.1: Areas and Distances
Math-101
Final Exam
Term-131
apply the idea to problem with large number of rectangles or more general problem (n rectangles)

10. 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 rectangles

11. Sec 5.1: Areas and Distances
Let’s apply the idea to the more general problem
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

12. 1 2 3 Sec 5.1: Areas and Distances Step Step Step partition
Riemann sum for ƒ on the interval [a, b].

13. Sec 5.1: Areas and Distances
Math-101
Final Exam
Term-131

14. Sec 5.1: Areas and Distances
Term-091

15. Sec 5.1: Areas and Distances
Term-103

16. Remark: Sec 5.1: Areas and Distances Actual area
How to get actual area ?

17. Remark: Sec 5.1: Areas and Distances Actual area Actual area
Big triangle _ Small triangel

19. Note: Area under a curve = limit of summation
Sec 5.1: Areas and Distances
Note: Area under a curve = limit of summation
Note:
Area under the curve equals 4

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 Distances
Example:
Use rectangles to estimate the area under the parabola
from 0 to 1

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

23. Sec 5.1: Areas and Distances
Example:
Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30-second time interval. We take speedometer readings every five seconds and record them in the following table:
Distance = velocity X time
assume that the velocity is constant in every 5-second interval
estimate for the
total distance traveled:
1135 ft