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
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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
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Midpoint and Trapezoidal Riemann Sums Rizzi Calc BC
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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 Lets add an initial
condition: The gorilla started from 150 meters. How far off the
ground was he at the end of 5 seconds?
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Warm Up AP Problem
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Motivation Right- and left-hand Riemann sums arent always
accurate Midpoint and Trapezoidal are more complex but can offer
more accurate estimations
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Midpoint Sum
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Midpoint Sum Graphical/Analytical
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Practice AP Problem Estimate the distance the train traveled
using a midpoint Riemann sum with 3 subintervals.
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Trapezoidal Sum Area of each interval is determined by finding
area of each trapezoid
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Trapezoidal Sum Graphical/Analytical
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Limits of Riemann Sums As we take more and more subintervals,
we get closer to the actual approximation of the area under the
curve.
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Limits of Riemann Sums Cont.
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Midpoint Sum - Numerical t0306090120150180
f(t)f(t)5.011.513.415.716.816.914.7