Absolute value of riemann sums

In summary, to prove that the difference between the upper and lower sums of the absolute value of a Riemann integrable function f is less than or equal to the difference between the upper and lower sums of f, we can use the reverse triangle inequality and the fact that f is Riemann integrable to show that |Sp|f| - sp|f| ≤ Spf - spf.
  • #1
missavvy
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Homework Statement



I'm trying to prove that
Sp|f| - sp|f| [tex]\leq[/tex] Spf - spf
Where P is a partition of [a,b] and f is function that is riemann integrable.

Homework Equations





The Attempt at a Solution



So I get to a point where M = supf(x) and m = inff(x)
then |M|(b-a) - |m|(b-a) = (|M|-|m|)(b-a) [tex]\leq[/tex] |M-m|(b-a)
From the reverse triangle inequality.
But I'm just confused since I don't want |M|-|m|[tex]\leq[/tex]|M-m|, I just want |M|-|m|[tex]\leq[/tex] (M-m)..

err.. help?
 
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  • #2


Hello! I am a scientist and I would like to offer some assistance with your proof. First, let's clarify the notation a bit. The notation "Sp|f|" and "sp|f|" typically refer to the upper and lower sums of a function f with respect to a partition P of [a,b]. So, your goal is to prove that the difference between the upper and lower sums of the absolute value of f is less than or equal to the difference between the upper and lower sums of f.

To start, let's recall the definition of upper and lower sums. The upper sum of f with respect to a partition P is defined as the sum of the largest values of f on each subinterval of P, multiplied by the length of that subinterval. Similarly, the lower sum is defined as the sum of the smallest values of f on each subinterval, multiplied by the length of the subinterval. So, we can rewrite your statement as:

Sp|f| - sp|f| ≤ Spf - spf

Now, let's focus on the left-hand side of the inequality. As you correctly stated, we can use the reverse triangle inequality to get:

Sp|f| - sp|f| = |Sp|f| - sp|f|| ≤ Sp|f| - sp|f|

So, we have reduced the problem to proving that |Sp|f| - sp|f| ≤ Spf - spf. To do this, we can use the fact that f is Riemann integrable. This means that for any given ε > 0, we can find a partition P' such that the difference between the upper and lower sums of f with respect to P' is less than ε. In other words:

Spf - spf ≤ ε

Now, let's consider the partition P' for the function |f|. Since |f| is also Riemann integrable, we can use the same argument to show that:

Sp|f| - sp|f| ≤ ε

Combining these two inequalities, we get:

Sp|f| - sp|f| ≤ ε ≤ Spf - spf

And we have proved the desired result. I hope this helps and good luck with your proof!
 

What is the definition of absolute value of Riemann sums?

The absolute value of Riemann sums is a mathematical concept used to determine the area under a curve by dividing the area into smaller, rectangular sections and adding them together. It represents the total distance between the curve and the x-axis, regardless of the direction of the curve.

How do you calculate the absolute value of Riemann sums?

To calculate the absolute value of Riemann sums, you need to first divide the area under the curve into smaller rectangles. Then, you need to calculate the area of each rectangle by multiplying its height (y-value) by its width (change in x-value). Finally, you add all of these areas together to get the total area under the curve.

What is the significance of the absolute value of Riemann sums?

The absolute value of Riemann sums is significant because it allows us to approximate the area under a curve, which is a common problem in mathematics and science. It also helps us to understand the behavior of functions and make predictions about their values at different points.

Are there any limitations to using the absolute value of Riemann sums?

Yes, there are some limitations to using the absolute value of Riemann sums. One limitation is that it can only be used for functions that are continuous and have a well-defined shape. Additionally, the accuracy of the approximation depends on the number of rectangles used, so using a larger number of rectangles can be more time-consuming and computationally intensive.

How is the absolute value of Riemann sums related to integration?

The absolute value of Riemann sums is closely related to integration, as it is one method for approximating the area under a curve. In fact, as the number of rectangles used in the calculation approaches infinity, the result becomes more and more accurate and approaches the exact value of the integral of the function.

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