Help with Apostol's "Calculus, vol. 1", Section 1.18

In summary: So, in a sense, s'(x) would be the LUB of s_n(x) for all n.In summary, Apostol proves two theorems about the area of a function's ordinate set and the area of the graph of a function. The first theorem, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem 1.11, deals with the area of the graph of the function.
  • #1
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In section 1.18 ("The area of an ordinate set expressed as an integral"), Apostol proves two theorems. the first, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem 1.11, deals with the area of the graph of the function of theorem 1.10. (I have attached two excrepts from Apostol's book, one per theorem.)

I am having problems understansing Apostol's logic in theorem 1.11 where he states:

"The argument used to prove Theorem 1.10 also shows that Q’ is measurable and that a(Q’) = a(Q)."

I don see how he could argue this, being that ##Q'=\{(x,y)|a \le x \le b, 0 \le y < f(x) \}## which implies that ##S \subseteq Q'## is not true for all step regions S (since S may contain a point of the graph of ##f(x)## which Q', by definition, can't).

Thanks in advanced for any help.


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  • #2
I think the method is the same, but the definition of S would have to change if a point in S was equal to f(x).
So for the measure of Q', you would have sets S' and T', but the result would be the same--that the area of a line (or curve) is zero.
  • #3
You can make the points ##(x, y)## in the set ##Q'## sufficiently close to the graph of ##y = f(x)##. So the areas are indeed equal.
  • #4
RUber, I see what you're saying, however the new step regions (S' and T') would produce functions bearing the following relationship with the function##f(x)##: ##s'(x) < f(x) \le t'(x)##. Which doesn't help since the definition of the integral requires "##\le##" for both step function inequalities.

Zondrina, The idea is making the of make the points equal since by definition of Q': ##Q \neq Q'##. So no matter how close I get too the points of ##f(x)##, I would not be able to put all points of both regions in a one-to-one correspondence in order to then argue by congruence of regions.
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  • #5
You can define a sequence s_n(x) such that the limit as n goes to infinity of s_n(x) = f(x). In this way, you will still satisfy the requirement that the intervals will converge on the true integral.
  • #6
I get what you're saying. The only problem is that Apostol still hasn't mentioned sequences. The only relevant thing I can think of that has been covered in the book till now is the Least Upper Bound Property of Numbers. Which would simply state that ##f(x)## is the supremum for ##Q'##

What is Apostol's "Calculus, vol. 1", Section 1.18?

Apostol's "Calculus, vol. 1", Section 1.18 is a section in the first volume of the textbook "Calculus" written by Tom M. Apostol. It covers the topic of integrals and their properties.

What is the difficulty level of Section 1.18?

The difficulty level of Section 1.18 can vary depending on the individual's understanding of calculus. However, it is generally considered to be a moderately difficult section.

What topics are covered in Section 1.18?

Section 1.18 covers the properties of integrals such as linearity, additivity, and monotonicity. It also discusses the fundamental theorem of calculus and its applications.

How can I get help with Section 1.18 of Apostol's "Calculus, vol. 1"?

If you are struggling with Section 1.18, you can seek help from your teacher, tutor, or classmates. You can also refer to online resources such as video tutorials, practice problems, and study guides.

Why is it important to understand integrals and their properties?

Integrals are a fundamental concept in calculus and have many real-world applications. Understanding their properties allows us to solve various problems in physics, engineering, economics, and other fields. It also helps in developing a deeper understanding of calculus as a whole.

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