1. More on Riemann Sums

2. We will go into more detail today on Riemann sums. Ex.1 Using the integralwe will approximate using sums with 4 subintervals.

3. Shown is L 4. x 1 3 5 7 9 Δx=2 Draw a number line for the limits of integration and find Δx.

4. Shown is L 4. x 1 3 5 7 9 Δx=2 Count off using the counting variable k, starting at 0. k 0 1 2 3 4

5. Shown is L 4. x 1 3 5 7 9 Δx=2 Now find f(x k ) for each k. f(x k ) 1.5 1 1.5 3 1.5 5 1.5 7 1.5 7 k 0 1 2 3 4

6. Shown is L 4. x 1 3 5 7 9 Δx=2 Now write in summation notation. f(x k ) 1.5 1 1.5 3 1.5 5 1.5 7 1.5 7 k 0 1 2 3 4 or

7. It isn’t necessary to write each step once you understand.

8. Shown is R 4. x 1 3 5 7 9 Δx=2 f(x k ) 1.5 1 1.5 3 1.5 5 1.5 7 1.5 7 k 0 1 2 3 4 We use the same number line as before.

9. Shown is R 4. x 1 3 5 7 9 Δx=2 Now write in summation notation. f(x k ) 1.5 1 1.5 3 1.5 5 1.5 7 1.5 7 k 0 1 2 3 4 or

11. Shown is M 4. x 1 3 5 7 9 Δx=2 f(x j ) 1.5 2 1.5 4 1.5 6 1.5 8 k 0 1 2 3 4 j 0 1 2 3 Identify midpoints in x, then go through the same process as before. x 2 4 6 8

12. Shown is M 4. f(x j ) 1.5 2 1.5 4 1.5 6 1.5 8 j 0 1 2 3 Write using summation notation. x 2 4 6 8 or

14. Compare to the true value of the integral.

15. The major point here is to begin to make the transition from the limits of integration in x, and writing in sigma notation. We have to introduce a counting variable, usually symbolized n, k, j, I, etc.