2. Jacob Bernoulli 1654 – 1705 Jacob Bernoulli 1654 – 1705 Jacob Bernoulli was a Swiss mathematician who became familiar with calculus through a correspondence with Gottfried Leibniz, then collaborated with his brother Johann on various applications, notably publishing papers on transcendental curves.

6. How about the area under the curve and above the x-axis? What does that area represent?

7. Right-Hand Rectangles Left-Hand Rectangles Midpoint Rectangles

8. Example: Find the area under the curve using Left-hand Rectangles. Approximate area:

9. We could also use a Right-hand Rectangles. Approximate area:

10. The other approach would be to use rectangles generated from the midpoint of each subinterval. The Midpoint Rectangles. Approximate area: In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

11. Approximate area: width of subinterval With 8 subintervals : What is the exact area?

14. This summation is called a Right Riemann Sum, it calculates the area under the curve.

17. Example: Find the area under the curve using a Riemann sum, over the interval [0,  ] with 4 subintervals, using right grid points. 7.584 Height of RectanglesBase of rectangles

18. Example: Find the area under the curve using a Riemann sum, over the interval [0,  ] with 20 subintervals, using right grid points. 7.984 Height of RectanglesBase of rectangles

19. This summation is called a Left Riemann Sum, it calculates the area under the curve.

20. This summation is called a Midpoint Riemann Sum, it calculates the area under the curve.

21. This summation is called a Riemann Sum, it calculates the area under the curve.

22. Example: http://mathworld.wolfram.com/RiemannSum.html