Name:
Date:
7.2a Notes on Riemann Sum
A train moves along a track at a steady rate of 75 miles per hour from 7 am to 9 am. What is
the total distance traveled by the train?
___________________________________________________________
Our goal today is to calculate the area under a curve. We do that by breaking the area (which
we cannot calculate) into rectangles (whose areas we can calculate).
Example 1: Given y  x 2 , what is the area under the curve from x = 0 to x = 4?
RAM is the Rectangular Approximation Method
LRAM uses the left hand values of y
RRAM uses the right hand values of y
MRAM uses the midpoint of L and R
Rectangular Approximation Method (RAM)
There are three rectangular approximation methods that we use:
LRAM:
RRAM:
MRAM:
Example 2: Given y  x 2  1 , calculate the area under the curve between x = -1 and x = 3
1. LRAM
2. MRAM
3. RRAM
Estimating Distance from Velocity:
Example 3: Suppose a car is moving with increasing velocity, and we measure the car’s velocity
every two seconds (see below)
Time (sec)
Velocity (ft/sec)
0
20
2
30
4
38
6
44
8
48
10
50
1. We can estimate the distance by assuming that the car’s speed is constant for each 2
second interval (using the velocity from the beginning of each interval).
2. We can also make a similar estimate using the velocity from the end of each interval.
3. What could be done to make the estimate better?
Time (sec)
Velocity (ft/sec)
0
20
1
26
2
30
3
35
4
38
5
42
6
44
7
46
8
48
9
49
10
50
Using one second intervals, the distance would be on the following interval
𝑓𝑒𝑒𝑡 ≤ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ≤
𝑓𝑒𝑒𝑡
By ________________ the size of the intervals, you can find better and better______________
Here are graphs of the actual velocity with the two-second and one-second estimates:
Example 4: A car comes to a stop six seconds after the drive applies the brakes. While the
brakes are on, the following velocities are recorded:
Time since
brakes applied
(sec)
Velocity (ft/sec)
0
2
4
6
88
45
16
0
What is the range of how far the car travels in those 6 seconds?
You can have a “negative” area under the curve. This is the ___________ area.
Example 5: Find the area under the following function.
7.2a Homework:
1. Given f ( x)  x 2 calculate the area under the curve using LRAM between x = 0 and x = 3.
2. Given f ( x)  x  1 calculate the area under the curve using LRAM and RRAM between [2,4]
3. Given f ( x)  ( x  2)2  1 calculate the area under the curve using LRAM and RRAM on [1,3].
4. Find the area under the function
5. Find the area under the function