5.1.notebook
January 12, 2016
5.1 Estimating Accumulation with Riemann Sums
Using the Rectangular Approximation Method (RAM) to estimate the area under a curve creates what is known as a Riemann sum.
To create a Riemann sum, we first decide how to partition the interval over which the sum will be calculated. This means, we have to decide how wide each rectangle will be. The rectangles can be of uniform width or their widths can be determined by the given data.
5.1.notebook
January 12, 2016
Next, a decision must be made about how to calculate the height(length) of each rectangle. For LRAM, the height of each rectangle is calculated using the output value at the left endpoint of the interval. Example:
input
0
1
2
3
4
5
6
output 10 20 30 40 50 60 70 For the subinterval 0 < input < 2, the height of the rectangle would be 10.
5.1.notebook
January 12, 2016
For RRAM, the height of each rectangle is calculated using the output value at the right endpoint of the interval. Example:
input
0
1
2
3
4
5
6
output 10 20 30 40 50 60 70 For the subinterval 0 < input < 2, the height of the rectangle would be 30.
5.1.notebook
January 12, 2016
For MRAM, the height of each rectangle is calculated using output value at the middle input value of the interval. Example:
input
0
1
2
3
4
5
6
output 10 20 30 40 50 60 70 For the subinterval 0 < input < 2, the height of the rectangle would be 20. Do we have enough information in our table to find the height of the rectangle on the interval from 0 < input < 1?
5.1.notebook
January 12, 2016
When calculating each Riemann sum, a table can be used to organize the process. Interval
Δt
f(t)
Area
However, to show work to support the Riemann sum calculation, you must write the sum of the products.
Δt(output) + Δt(output) + Δt(output) = sum
5.1.notebook
January 12, 2016
Example: As a research student you are walking along the bank of a tidal river watching the incoming tide carry a bottle of water upstream. In order to study the affects of the current in the river, you drop a measuring tool in and record the velocity of the flow every 5 minutes for a half hour. The results are shown in the table below. Time (min)
0
Velocity (m/min)
1
5
10
15
20
25
30
1.2 1.7 2.0 1.8 1.6 1.4
5.1.notebook
January 12, 2016
Time (min)
0
Velocity (m/min)
1
5
10
15
20
25
30
1.2 1.7 2.0 1.8 1.6 1.4
a) Estimate the distance the bottle has traveled in thirty minutes using a left Riemann sum with three subintervals of equal length.
5.1.notebook
January 12, 2016
a) Estimate the distance the bottle has traveled in thirty minutes using a left Riemann sum with six subintervals of equal length. This table is a graphic organizer.It does not need to be made, but I use it to organize my thought process to calculate the sum.
Interval
Δt
v(t)
Area
0 < t < 10
10
1
10
10 < t < 20
10
1.7
17
20 < t < 30
10
1.8
18
10(1) + 10(1.7) + 10(1.8) = 45 miles
5.1.notebook
January 12, 2016
Time (min)
0
Velocity (m/min)
1
5
10
15
20
25
30
1.2 1.7 2.0 1.8 1.6 1.4
b) Estimate the distance the bottle has traveled in thirty minutes using a right Riemann sum with three subintervals of equal length.
5.1.notebook
January 12, 2016
b) Estimate the distance the bottle has traveled in thirty minutes using a right Riemann sum with six subintervals of equal length. This table is a graphic organizer.It does not need to be made, but I use it to organize my thought process to calculate the sum.
Interval
Δt
v(t)
Area
0 < t < 10
10
1.7
17
10 < t < 20
10
1.8
18
20 < t < 30
10
1.4
14
10(1.7) + 10(1.8) + 10(1.4) = 49 miles
5.1.notebook
January 12, 2016
Time (min)
0
Velocity (m/min)
1
5
10
15
20
25
30
1.2 1.7 2.0 1.8 1.6 1.4
c) Estimate the distance the bottle has traveled in thirty minutes using a midpoint Riemann sum with three subintervals of equal length.
5.1.notebook
January 12, 2016
c) Estimate the distance the bottle has traveled in thirty minutes using a midpoint Riemann sum with three subintervals of equal length. This table is a graphic organizer.It does not need to be made, but I use it to organize my thought process to calculate the sum.
Interval
Δt
v(t)
Area
0 < t < 10
10
1.2
12
10 < t < 20
10
2.0
20
20 < t < 30
10
1.6
16
10(1.2) + 10(2.0) + 10(1.6) = 48 miles
