Upper and Lower sums & Riemann sums

[Call my name]

Homework Statement

Homework Equations

The Attempt at a Solution

I have attachments that can answer the above template, and please look at the attachments if you are trying to help me.

I have two questions regarding upper and lower sums & Riemann sums.

So, the attachments 1 & 2 are example 5 and 6 from the book that I am using. I have underlined in red where I cannot understand how the examples are solved.

Where do these two underlined inequalities come from? How can I find these inequalities (method/approach)?

I want to make sure that I understand these and practice some other questions of the same type, so if you happen to have some question(s) of this type, please inform me.

The third attachment, 3 is the continuation of explanation of Riemann sums, and then the book starts by llPll, which I do not understand, and end the chapter with some limit, which I also do not understand.

My name is Ace, and I will say hello now since you've read through my lines. Please clarify these questions if you have a clue and can explain in a good way. Thanks.

 

[mighty2000]

Dear Ace,

Thank you for reaching out for help with your questions about upper and lower sums and Riemann sums. I am happy to assist you in understanding these concepts.

To answer your first question about the underlined inequalities in examples 5 and 6, these inequalities come from the definition of upper and lower sums. In general, the upper sum is the sum of the largest rectangles that fit under the curve, while the lower sum is the sum of the smallest rectangles that fit above the curve. In these examples, the inequalities are used to determine the height of each rectangle, which is determined by the function value at each partition point. To find these inequalities, you can use the same method/approach as shown in the examples, which is to evaluate the function at each partition point and compare it to the previous value to determine if it is larger or smaller.

As for your second question about llPll, this is the notation for the absolute value of P, the partition of the interval [a,b]. In other words, it represents the length of the interval [a,b] that is being divided into smaller subintervals. This is used in the limit at the end of the chapter to show that as the partition becomes finer (i.e. the length of the subintervals approaches 0), the upper and lower sums approach the same value, which is the Riemann sum.

I hope this helps clarify your questions and understanding of upper and lower sums and Riemann sums. I would suggest practicing with more examples and problems to solidify your understanding. If you have any further questions, please do not hesitate to ask.

Best of luck with your studies!