1. Converting a Definite Integral to a limit of a Riemann Sum and converting a limit of a Riemann Sum to a Definite Integral
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8-R

2. Riemann Sum approximation for the Area of Region
If a region is bounded above by f(x) and below by the x-axis at all points of the interval [a,b], then the area of the region bounded by x = a and x = b
using Right-hand rectangles is
Height of Rectangle
Width of Rectangle

3. Some Curve

4. Area Under a Curve
Find the area of the region bounded by y = f(x), the x-axis, x = a, and x = b.
Approximate the area by creating rectangles of equal width whose endpoints are on f(x).
n = # of rectangles
i = interval
Each left endpoint is on f(x)
Each right endpoint is on f(x)
Each method is called a Riemann Sum.

5. The Limit Definition for finding the area under a curve using a right hand Riemann Sum
Formula
Height of the rectangle
Width of the rectangle

6. Limit of a Riemann Sum
The definite integral of a continuous function f over the interval [a, b] denoted by is the limit of Riemann sums as the widths of the subintervals approach 0. That is Where is the value in the ith subinterval, and Is the width of the ith subinterval. And n is the number of subintervals

7. 1) Express the definite integral as the limit of a Riemann Sum

8. 2) Express the definite integral as the limit of a Riemann Sum

9. 3) Express the limit of a Riemann Sum as a definite integral

10. 4) Express the limit of a Riemann Sum as a definite integral

11. 5) Express the limit of a Riemann Sum as a definite integral

12. 6) Express the limit of a Riemann Sum as a definite integral

13. Summation Formulas
Proof of these formulas are found in Appendix A of our book

14. 7) Find the area beneath (above the x-axis) in the interval [1,3] using the limit definition of a Riemann Sum

15. Summation Formula

16. Home Work
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Worksheet 8-R