Left and RightHand Riemann Sums Rizzi Calc BC

Left and Right-Hand Riemann Sums Rizzi – Calc BC

The Great Gorilla Jump

The Great Gorilla Jump 20 18 16 Velocity (m/s) 14 12 10 8 6 4 2 0 0 1 2 3 Time (s) 4 5 6

Left-Hand Riemann Sum

Right-Hand Riemann Sum

Over/Under Estimates

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

Practice AP Problem t (hours) 0 The rate of fuel consumption 30 (in gallons per minute) recorded 40 during a plane flight is given by a 50 twice-differentiable function R of 70 time t, in minutes. 90 R(t) 20 30 40 55 65 70 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?

Midpoint and Trapezoidal Riemann Sums Rizzi – Calc BC

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?

Warm Up AP Problem E

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

Midpoint Sum

Midpoint Sum – Graphical/Analytical •

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

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

Trapezoidal Sum – Graphical/Analytical •

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

Limits of Riemann Sums Cont.

Midpoint Sum - Numerical t 0 f(t) 5. 0 30 60 90 120 150 180 11. 5 13. 4 15. 7 16. 8 16. 9 14. 7