Upper and Lower Darboux Sum Inequality

In summary, when comparing partitions of a bounded function on a closed interval, if the finer partition is a subset of the courser partition, then the lower sum of the finer partition will be less than or equal to the lower sum of the courser partition and the upper sum of the courser partition will be less than or equal to the upper sum of the finer partition. This is due to the fact that the finer partition will be closer to the limit in both upper and lower sums, with the supremums of the courser partition being applied to smaller intervals.
  • #1
Magnetons
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TL;DR Summary
L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P)
Lemma
Let f be a bounded function on [a,b]. If P & Q are partitions of [a,b] and P ##\subseteq## Q , then

L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P) .
Question is "How can P have bigger upper darboux sum than Q while it is a subset of Q"
 
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  • #2
The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller ##\Delta x##s.
 
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  • #3
FactChecker said:
The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller ##\Delta x##s.
 

FAQ: Upper and Lower Darboux Sum Inequality

What is the Upper Darboux Sum?

The Upper Darboux Sum is a method of approximating the area under a curve by dividing the interval into subintervals and using the supremum (least upper bound) of the function values on each subinterval to create rectangles. The sum of the areas of these rectangles gives an upper approximation of the integral of the function over that interval.

What is the Lower Darboux Sum?

The Lower Darboux Sum is similar to the Upper Darboux Sum but uses the infimum (greatest lower bound) of the function values on each subinterval to create rectangles. The sum of the areas of these rectangles provides a lower approximation of the integral of the function over the specified interval.

How do Upper and Lower Darboux Sums relate to the concept of Riemann integrability?

The Upper and Lower Darboux Sums are fundamental in determining whether a function is Riemann integrable. A function is Riemann integrable on an interval if the difference between the Upper and Lower Darboux Sums approaches zero as the partition of the interval becomes finer. Specifically, if the limit of the Upper Darboux Sum minus the Lower Darboux Sum is zero, the function is integrable over that interval.

What is the Darboux Sum Inequality?

The Darboux Sum Inequality states that for any partition of the interval, the Lower Darboux Sum is less than or equal to the Upper Darboux Sum. Mathematically, if L(P) is the Lower Darboux Sum and U(P) is the Upper Darboux Sum for a partition P, then L(P) ≤ U(P). This inequality is crucial for showing the boundedness of the integral estimates.

How can we use Upper and Lower Darboux Sums to estimate the area under a curve?

To estimate the area under a curve using Upper and Lower Darboux Sums, first, choose a partition of the interval over which you want to integrate the function. Then, calculate the Upper Darboux Sum by taking the supremum of the function on each subinterval and multiplying by the width of the subintervals. Similarly, calculate the Lower Darboux Sum using the infimum. The area can be approximated by these sums, with the Upper Sum providing an overestimate and the Lower Sum providing an underestimate.

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