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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 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. t (hours) R(t) 0 30 40 50 20 30 40 55 70 90 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 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 • Approximate the area under the curve = 2 + 1 on the interval [0, 8] using a midpoint sum with 4 equal subintervals. 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 • Estimate 4 1 + −1 using the trapzezoidal rule with n = 5 subintervals. 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 14.7 180 () 0 Estimate by using a midpoint sum with three subintervals