1. To find the area under the curve Warm-Up: Graph

2. Area under a curve for [0, 3]  The area between the x-axis and the function Warm-up What is the area under the function on the interval [0, 3]?

3. Find the area of the function from [0, 5]

4. Find the area under the function Problem???

5. Riemann Sums  Approximates the area under the curve using rectangles  Three different orientations for the rectangles left

6. Riemann Sums  Approximates the area under the curve using rectangles  Three different orientations for the rectangles left middle

7. Riemann Sums  Approximates the area under the curve using rectangles  Three different orientations for the rectangles left middle right

8. Riemann Sums  Sum the areas of all rectangle -the height of each rectangle is f(x) -the width of each rectangle is dx

9. Use Riemann sums to find the area under the curve

10. Left Riemann Sums Let dx = 0.5

11. Left Riemann Sum

12. Middle Riemann Sum

14. Right Riemann Sums

16. Right vs Left Riemann Sums

17. Determine the following  Given a function, an interval and dx Find: 1. Sketch the function and the approximating rectangles using the midpoints 2. Approximate the area under the curve

18. On the interval [0, 4] dx = 1 1. Sketch

19. 2. Area