1. A Method For Approximating the Areas of Irregular Regions
Riemann Sums
A Method For Approximating the Areas of Irregular Regions

2. The Necessity for Approximation
In previous courses, you’ve learned how to find the areas of regular geometric shapes using various formulas…

3. The Necessity for Approximation
However, when we have shapes like these, there are no nice, neat formulas with which to calculate their area…

4. The Necessity for Approximation
To address this issue, we use a large number of small rectangles to approximate the area of one of these irregular regions. We may choose to use rectangles with different widths or rectangles with the same width.

5. The Necessity for Approximation
Things to Remember
When we know a function, it is best to approximate the area using rectangles with the same width.
When we only know certain points of the function, we will let those points dictate the widths of the rectangles that we use.

6. The BIG Unanswered Question???
We’ve talked about the widths of the rectangles that we will use for approximation, but how will we decide what the heights of these rectangles will be?

7. Determining the Heights of our Rectangles
Basically, unless specified, we can use any point on the function in the given interval for the height of a rectangle. However, we typically choose to use one of 3 points:
The Left Endpoint
The Right Endpoint
The Midpoint

8. BEWARE OF SCARY NOTATION
Caution!
BEWARE OF
SCARY NOTATION
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9. In Left Riemann sum, the left-side sample of the function is used as the height of the individual rectangle.

10. In Right Riemann sum, the right-side sample of the function is used as the height of the individual rectangle.

11. In the Midpoint Rule, the sample at the middle of the subinterval is used as the height of the individual rectangle.

12. In English Please!!!
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13. Left Hand Rectangular Approximation Method
As its name indicates, in the Left Hand Rectangular Approximation Method (LRAM), we will use the value of the function at the left endpoint to determine the heights of the rectangles.

14. Left Hand Rectangular Approximation Method

15. Left Hand Rectangular Approximation Method

16. Right Hand Rectangular Approximation Method
As its name indicates, in the Right Hand Rectangular Approximation Method (RRAM), we will use the value of the function at the right endpoint to determine the heights of the rectangles.

17. Right Hand Rectangular Approximation Method

18. Right Hand Rectangular Approximation Method

19. Midpoint Rectangular Approximation Method
As its name indicates, in the Midpoint Rectangular Approximation Method (MRAM), we will use the value of the function at the midpoint to determine the heights of the rectangles.

20. Midpoint Rectangular Approximation Method

21. Midpoint Rectangular Approximation Method

22. Approximations from Numeric Data
Time
Speed 3 8 5 14 11 7 15 18
These methods for approximation are most useful when we don’t actually know a function, but where we have several numerical data points that have been collected. In this situation, the widths of our rectangles will be determined for us (and may not all be the same).

23. Approximations from Numeric Data
To find the distance traveled in this situation, we can choose to use either LRAM or RRAM (but not MRAM since we don’t know the values at the midpoints. However, unlike our previous examples, the widths of our rectangles will all be different.
Time
Speed 3 8 5 14 11 7 15 18

24. Approximations from Numeric Data
To find LRAM, we will use the function value at the left endpoint and the varying widths of rectangles.
Time
Speed 3 8 5 14 11 7 15 18

25. Approximations from Numeric Data
Time
Speed 3 8 5 14 11 7 15 18
To find RRAM, we will use the function value at the right endpoint and the varying widths of rectangles.

26. What if I HATE Rectangles?? Can I use another shape?
More importantly, is there a shape that gives a better approximation?
Let’s look at an example that explores a new approximation
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27. Actual area under curve:

28. Left-hand rectangular approximation:
Approximate area:
(too low)

29. Right-hand rectangular approximation:
Approximate area:
(too high)

30. Averaging the two:
1.25% error
(too high)

31. Averaging right and left rectangles gives us trapezoids:

32. (still too high)

33. Trapezoidal Rule: ( h = width of subinterval )
This gives us a better approximation than either left or right rectangles.

34. Compare this with the Midpoint Rule:
Approximate area:
0.625% error
(too low)
The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

35. Ahhh! Oooh! Wow! Trapezoidal Rule: 1.25% error (too high)
Midpoint Rule:
(too low)
0.625% error
Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction.
Ahhh!
If we use a weighted average:
This is the exact answer!
Oooh!
Wow!

36. We got properties for indefinite integrals….
What now??
Do the same rules apply?
What if the curve goes below the axis?
Let’s do some examples!!
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37. Negative Values
If f (x) is positive for some values of x on [a, b] and negative for others, then the definite integral symbol
y = f (x) B a
represents the cumulative sum of the signed areas between the graph of f (x) and the x axis, where areas above are positive and areas below negative. A b

38. Examples
Calculate the definite integrals by referring to the figure with the indicated areas.
Area A = 3.5
Area B = 12 B a A b c
y = f (x)

39. Definite Integral Properties

40. Examples
Assume we know that
Then A) B)

41. Examples (continued) C) D) E)