(revised+re-post)Upper and Lower sums & Riemann sums

In summary, the conversation discusses the concept of inequalities in relation to Riemann sums and integrals. The speaker asks about the origin of these inequalities and how they can be approached. They also mention difficulties understanding the sign "llPll" and how it relates to upper and lower sums. They express a desire for clarification and understanding in this topic.
  • #1
Call my name
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If you look at the above, I have underlined the problem that I am having.

So, my first question is, where are these inequalities coming from? If you do have other questions involving such approach, please show me.

My other question is from the explanation of Riemann sum, I do not understand the sign "llPll
" thing and I am having trouble understanding Riemann sum and how it really is related to the upper and lower sums.

Thank you for your attention.
 
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  • #3
you can write those reasonable easy with tex, click on it below

1)
[tex] x_{i-1} \leq \frac{x_{i-1} +x_{i}}{2} \leq x_{i} [/tex]

comes pretty easy as by defintion of your partition
[tex] x_{i-1} \leq x_{i} [/tex]
and probably more actually
[tex] x_{i-1} < x_{i} [/tex]

split them into two equalities
[tex] x_{i-1} \leq \frac{x_{i-1} +x_{i}}{2} [/tex]
[tex] \frac{x_{i-1} +x_{i}}{2} \leq x_{i} [/tex]

multiplying everything by 2 and subtract something & it should be easy to see

2) same thing as before, and teh fact that in this case you know [itex] x_i \geq 1[/itex] , so [itex] x_{i-1}^2 <x_i^2 [/itex]

3) the last one is the definition of the integral as the limit of the sum when every partition appraches zero
 
  • #4
I understand that these inequalities actually do 'work',

but what I do not understand is how you 'approach' these questions.

Like, I understand the concept of upper and lower sums, but how do I come up with the inequalities in the first place?

Do I need to first solve the integral by using the fundamental theorem, and then try to make it look that way by coming up with some inequalities?

How do I in the first place just go 'boom' and come up with the 2nd page's inequality?

Thanks for your attention and hopefully somebody will answer..
 

1. What is the difference between upper and lower sums and Riemann sums?

The upper and lower sums are approximations of the area under a curve using rectangles. The upper sum uses the maximum value of the function on each subinterval, while the lower sum uses the minimum value. Riemann sums are a generalization of the upper and lower sums, where the number of rectangles used approaches infinity, resulting in a more accurate approximation of the area.

2. How do you calculate the upper and lower sums of a function?

To calculate the upper and lower sums, you first divide the interval of interest into subintervals of equal width. Then, for each subinterval, you find the maximum and minimum values of the function within that interval. Finally, you multiply the width of the subinterval by the maximum or minimum value, depending on whether you are calculating the upper or lower sum, and sum up all the resulting values.

3. What is the significance of Riemann sums in calculus?

Riemann sums are important in calculus because they provide a way to approximate the area under a curve, which can be used to find the definite integral of a function. This allows us to solve problems involving finding the area under a curve, such as calculating the work done by a variable force or finding the distance traveled by an object with a varying velocity.

4. How do you improve the accuracy of Riemann sums?

The accuracy of Riemann sums can be improved by increasing the number of rectangles used in the approximation. This can be done by decreasing the width of the subintervals, resulting in a more accurate approximation of the area under the curve. Another way to improve accuracy is by using more advanced techniques, such as the trapezoidal rule or Simpson's rule, which use more sophisticated shapes, such as trapezoids or parabolas, to approximate the area.

5. Are Riemann sums always exact?

No, Riemann sums are not always exact. They are only exact when the function being integrated is a polynomial with a degree equal to or less than the number of rectangles used in the approximation. Otherwise, Riemann sums provide an approximation of the area under the curve, with the accuracy increasing as the number of rectangles used approaches infinity.

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