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 𝑦 = 𝑥 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 + 𝑥 𝑑𝑥
−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
𝑓(𝑡)
0
Estimate
by using a midpoint
sum with three subintervals