Upper and lower Riemann sums

In summary, the upper Riemann sum of a Riemann integrable function f on an interval [a,b] is defined as U(f,P) = \sum_{i=1}^n M_i (x_i - x_{i-1}), where M_i = \sup\limits_{x\in [x_{i-1},x_i]} f(x). This definition uses \sup instead of \max, as there may be instances where the maximum value of f is not actually attained within the interval [x_{i-1},x_i]. One example of this is the function y=x when x\neq 1, y=0 when x=1 on the interval [0,1].
  • #1
AxiomOfChoice
533
1
Let [itex]f[/itex] be a Riemann integrable function defined on an interval [itex][a,b][/itex], and let [itex]P = \{a = x_0 < x_1 < \ldots < x_n = b\}[/itex] be a partition of [itex][a,b][/itex]. I don't understand why the definition of (say) the upper Riemann sum of [itex]f[/itex] associated with [itex]P[/itex] is always given as
[tex]
U(f,P) = \sum_{i=1}^n M_i (x_i - x_{i-1}),
[/tex]

where

[tex]
M_i = \sup\limits_{x\in [x_{i-1},x_i]} f(x).
[/tex]

Why does it need to be [itex]\sup[/itex]? Why can't it just be [itex]\max[/itex]? I can't think of an instance where [itex]\max[/itex] wouldn't be suitable; that is, I can't think of a Riemann integrable function that doesn't actually attain its max somewhere on any given [itex][x_{i-1},x_i][/itex]. Can someone provide an example for me?
 
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  • #2
[itex]y=x[/itex] when [itex]x\neq 1[/itex], [itex]y=0[/itex] when [itex]x=1[/itex] on interval [itex][0,1][/itex]. Take interval [itex][x_{n-1},x_n][/itex] from the partition.
 

What is the definition of Upper and Lower Riemann sums?

The Upper and Lower Riemann sums are methods used in calculus to approximate the area under a curve. They divide the area into smaller rectangles and calculate the sum of their areas.

What is the difference between Upper and Lower Riemann sums?

The main difference between Upper and Lower Riemann sums is the way they choose the height of the rectangles. In Upper Riemann sums, the height is chosen to be the maximum value of the function in a given interval, while in Lower Riemann sums, the height is chosen to be the minimum value.

What is the purpose of using Upper and Lower Riemann sums?

The purpose of using Upper and Lower Riemann sums is to approximate the area under a curve when it is not possible to find the exact value using traditional methods. They give a close estimate of the area and become more accurate as the number of rectangles used increases.

How do you calculate Upper and Lower Riemann sums?

To calculate Upper and Lower Riemann sums, you need to divide the given interval into smaller subintervals, choose the height of the rectangles, and then calculate the sum of their areas. The Upper Riemann sum is calculated by multiplying the width of the subinterval by the maximum value of the function in that subinterval, while the Lower Riemann sum is calculated by multiplying the width of the subinterval by the minimum value.

What is the significance of Upper and Lower Riemann sums in calculus?

Upper and Lower Riemann sums play a crucial role in calculus as they are used to approximate the area under a curve, which is an essential concept in finding the definite integral of a function. They also help in understanding the behavior of a function by breaking it down into smaller parts and analyzing the area under each part.

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