Math 1325
Lesson 16
Area and Riemann Sums
To find the area for the special nice shapes, we use the following formulas.
Then how do we find the area of the regions below? This is the other big topic in calculus.
We estimate the area of the regions above using the sum of rectangle areas.
Lesson 16 – Area and Riemann Sums
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Left Endpoints Estimation
Lesson 16 – Area and Riemann Sums
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Right Endpoints Estimation
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Midpoints Estimation
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The width of each rectangle on the interval [a, b] divided into n subintervals: x 
The height of the rectangles using 𝒙𝟏 , 𝒙𝟐 , ⋯ , 𝒙𝒏 :
ba
n
𝑓(𝑥1 ), 𝑓(𝑥2 ), ⋯ , 𝑓(𝑥𝑛 )
If 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 are left endpoints, then this method is called left endpoints estimation.
If 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 are right endpoints, then this method is called right endpoints estimation.
If 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 are midpoints, then this method is called midpoints estimation.
Estimation of the area is the sum of rectangles’ area:
𝐴 ≈ ∆𝑥 ∙ 𝑓(𝑥1 ) + ∆𝑥 ∙ 𝑓(𝑥2 ) + ⋯ + ∆𝑥 ∙ 𝑓(𝑥𝑛 ) = [𝑓(𝑥1 ) + 𝑓(𝑥2 ) + ⋯ + 𝑓(𝑥𝑛 )] ∙ ∆𝑥
 f  x1   f  x2  
Actual area: A  lim
n  
 f  xn    x = lim ∑𝑛𝑘=1 𝑓(𝑥𝑘 ) ∙ ∆𝑥
𝑛→∞
The process we are using to approximate the area under the curve is called “finding a
Riemann sum.” These sums are named after the German mathematician who developed them.
Example 1: Let f ( x )  2 x  1 . Approximate the area under the curve, using 4
subdivisions, on the interval [0, 2] using left endpoints.
2
Now suppose the function, interval and/or subdivisions we wish to work with are not so nice.
You would not want to work this type of problem by hand. We can work Riemann sum
problems using GeoGebra.
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RectangleSum[<Function>, <Start x-Value>, <End x-Value>, <Number of rectangles>,
<Position for rectangle start>]: the area between the x-axis and the graph of the function
“Position of rectangle start”: 0 corresponds to left endpoints
0.5 corresponds to midpoints
1 corresponds to right endpoints.
Example 2: Approximate the area between the x-axis and the graph of
f ( x)  0.3x3  0.807 x 2  3.252 x  6.717 on [-2.82, 1.33] with:
Enter the function in GGB.
a. 50 rectangles and midpoints.
Command:
Answer:
b. 10 rectangles and right endpoints.
Command:
Answer:
Example 3: Approximate the area between the x-axis and the function
f ( x) 
15
on [0.075, 8.21] using 12 rectangles and left endpoints.
0.083x 2  19.17 x  1
Enter the function in GGB.
Command:
Lesson 16 – Area and Riemann Sums
Answer:
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Upper Sum Estimation
Lower Sum Estimation
The upper sum uses the biggest y value on each interval as the height of the rectangle and the
lower sum uses the smallest y value on each interval as the height of the rectangle.
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Upper and Lower Sums Using GeoGebra
UpperSum[<Function>,<Start x-Value>,<End x-Value>,<Number of Rectangles>]
LowerSum[<Function>,<Start x-Value>,<End x-Value>,<Number of Rectangles>]
Example 4: Use GeoGebra to find the upper sum and the lower sum for f  x  
1 x
e  2 on
2
the interval [-3, 5] using 35 rectangles. Enter the function in GGB.
Command:
Upper Sum Answer:
Command:
Lower Sum Answer:
The Definite Integral
Let f be defined on [a, b].
If lim [𝑓(𝑥1 ) + 𝑓(𝑥2 ) + ⋯ + 𝑓(𝑥𝑛 )] ∙ ∆𝑥 = lim ∑𝑛
𝑘=1 𝑓(𝑥𝑘 ) ∙ ∆𝑥 exists for all choices of
𝑛→∞
𝑛→∞
representative points in the n subintervals of [a, b] of equal width x 
ba
, then this limit is
n
called the definite integral of f from a to b. The definite integral is noted by

b
a
f ( x)dx = lim ∑𝑛𝑘=1 𝑓(𝑥𝑘 ) ∙ ∆𝑥. The number a is called the lower limit of integration
𝑛→∞
and the number b is called the upper limit of integration.

b
a
f ( x)dx give us the area of the
region between the x-axis and the function f.
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