1. Left and Right-Hand Riemann Sums Rizzi – Calc BC

2. The Great Gorilla Jump

4. Left-Hand Riemann Sum

5. Right-Hand Riemann Sum

6. Over/Under Estimates

7. Riemann Sums Summary Way to look at accumulated rates of change over an interval Area under a velocity curve looks at how the accumulated rates of change of velocity affect position Area under an acceleration curve looks at how the accumulated rates of change of acceleration affect velocity

8. Practice AP Problem The rate of fuel consumption (in gallons per minute) recorded during a plane flight is given by a twice-differentiable function R of time t, in minutes. 1.Approximate the value of the total fuel consumption using a left-hand Riemann sum with the five subintervals listed in the table above. 2.Over or under estimation? Why? t (hours)R(t) 020 30 40 5055 7065 9070

9. Midpoint and Trapezoidal Riemann Sums Rizzi – Calc BC

10. Area Under Curve Review In the gorilla problem yesterday, area under the curve referred to the total distance the gorilla fell This is an accumulated rate of change Let’s add an initial condition: The gorilla started from 150 meters. How far off the ground was he at the end of 5 seconds?

11. Warm Up AP Problem

12. Motivation Right- and left-hand Riemann sums aren’t always accurate Midpoint and Trapezoidal are more complex but can offer more accurate estimations

13. Midpoint Sum

14. Midpoint Sum – Graphical/Analytical

15. Practice AP Problem Estimate the distance the train traveled using a midpoint Riemann sum with 3 subintervals.

16. Trapezoidal Sum Area of each interval is determined by finding area of each trapezoid

17. Trapezoidal Sum – Graphical/Analytical

18. Limits of Riemann Sums As we take more and more subintervals, we get closer to the actual approximation of the area under the curve.

19. Limits of Riemann Sums Cont.

20. Midpoint Sum - Numerical t0306090120150180 f(t)f(t)5.011.513.415.716.816.914.7