Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

In summary: To see this, let $P_{\epsilon}$ be a partition such that $F(a)-F(b) \leq P_{\epsilon}$. Then by (6) there exists a number $\epsilon\gt 0$ such that $sup_{\mathcal{L}(\mathcal{P},f)\}-\epsilon=F(b)-F(a)$ for all $f \in \mathcal{P}$. Since $\mathcal{L}(P_{\epsilon},f) \leq F(a)-F(b)$, $\mathcal{L}(P_{\epsilon},f)$
  • #1
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I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of The Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

Stoll's statement of Theorem 6.3.2 and its proof reads as follows:View attachment 3952
View attachment 3953
In the above proof we read:

Since

\(\displaystyle \mathscr{L}( \mathscr{P} , f ) \le \sum_{i = 1}^n f(t_i) \Delta x_i \le \mathscr{U}( \mathscr{P} , f )\)

we have

\(\displaystyle \underline{\int^b_a}f(x) dx \le F(b) - F(a) \le \overline{\int^b_a} f(x) dx \)My question is as follows: How do we know this is true? or Why exactly is this the case?

Can someone explain?
To restate my question:Since \(\displaystyle \underline{\int^b_a} f(x) dx\) is the supremum of \(\displaystyle \mathscr{L}( \mathscr{P} , f )\) over all partitions \(\displaystyle \mathscr{P}\) how do we know that it is less than F(b) - F(a)? ... ...... ... that is, is it possible for the quantity \(\displaystyle \mathscr{L}( \mathscr{P} , f ) \)to be less than \(\displaystyle F(b) - F(a)\) for all partitions \(\displaystyle \mathscr{P}\) but for the supremum over all partitions to fail to be less than \(\displaystyle F(b) - F(a)\)? ... ...
Hmm ... ... reflecting ... ... beginning to think this is not possible ... but how would you frame a rigorous symbolic argument of this?

Can someone please help?

Peter
 
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  • #2
Peter said:
I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of The Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

Stoll's statement of Theorem 6.3.2 and its proof reads as follows:

In the above proof we read:

Since

\(\displaystyle \mathscr{L}( \mathscr{P} , f ) \le \sum_{i = 1}^n f(t_i) \Delta x_i \le \mathscr{U}( \mathscr{P} , f )\)

we have

\(\displaystyle \underline{\int^b_a}f(x) dx \le F(b) - F(a) \le \overline{\int^b_a} f(x) dx \)My question is as follows: How do we know this is true? or Why exactly is this the case?

Can someone explain?
To restate my question:Since \(\displaystyle \underline{\int^b_a} f(x) dx\) is the supremum of \(\displaystyle \mathscr{L}( \mathscr{P} , f )\) over all partitions \(\displaystyle \mathscr{P}\) how do we know that it is less than F(b) - F(a)? ... ...... ... that is, is it possible for the quantity \(\displaystyle \mathscr{L}( \mathscr{P} , f ) \)to be less than \(\displaystyle F(b) - F(a)\) for all partitions \(\displaystyle \mathscr{P}\) but for the supremum over all partitions to fail to be less than \(\displaystyle F(b) - F(a)\)? ... ...
Hmm ... ... reflecting ... ... beginning to think this is not possible ... but how would you frame a rigorous symbolic argument of this?

Can someone please help?

Peter
After a little reflection, I will try to answer my own question ...We have that:

\(\displaystyle \mathscr{L}( \mathscr{P} , f ) \le F(b) - F(a)\) for any (i.e. every) partition \(\displaystyle \mathscr{P}\) ... ... ... (1)
We want to show that it follows from (1) that:\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} \le F(b) - F(a) \) ... ... ... (2)
Now proceed with proof:Suppose (2) does not follow ... ... then we have:\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} \gt F(b) - F(a)\) ... ... ... (3)Then there exists an \(\displaystyle \epsilon \gt 0\) such that:

\(\displaystyle sup \{ \mathscr{L}( \mathscr{P} , f ) \} - \epsilon = F(b) - F(a)\) ... ... ... (4)Then we have that:

\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} - \frac{\epsilon}{2} \gt F(b) - F(a)\) ... ... ... (5)But from Lemma 1.2.10 in Pons: Real Analysis for Undergraduates (see below and also see my reply to Fallen Angel in my post:http://mathhelpboards.com/analysis-50/riemann-criterion-integrability-stoll-theorem-6-17-a-14292.html ) we have that there is a partition \(\displaystyle \mathscr{P}_{ \frac{\epsilon}{2} }\) such that:\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} - \frac{\epsilon}{2} \lt \mathscr{L} ( \mathscr{P}_{ \frac{\epsilon}{2} } , f ) \) ... ... ... (6)
Now (5), (6) imply that:\(\displaystyle \mathscr{L} ( \mathscr{P}_{ \frac{\epsilon}{2} } , f ) \gt F(b) - F(a) \)

which is a contradiction ... ...
I am somewhat unsure of the correctness and reasonableness of my analysis ... ... so ... Can someone please critique the above analysis, pointing out any errors or misconceptions ...

Would appreciate some help ... ...

Peter***EDIT***

In the above post I mentioned Pons' Lemma 1.2.10 which I referred to and provided in a previous post.

For the convenience of MHB members reading this post I have decided to re-provide it here, as follows:
View attachment 3958
https://www.physicsforums.com/attachments/3959
 
Last edited:
  • #3
Hi Peter,

Your argument is correct but there is a direct way of doing this.

$\mathcal{L}(\mathcal{P},f)\leq F(a)-F(b)$ for any partition $\mathcal{P}$ means that $F(a)-F(b)$ is an upper bound for $\{\mathcal{L}(\mathcal{P},f)\}$ and the supremum is always less or equal than any upper bound.
 

Related to Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a theorem in calculus that establishes a relationship between differentiation and integration. It states that the integral of a function f(x) over an interval [a, b] can be calculated by finding the antiderivative of f(x) and evaluating it at the endpoints a and b.

What is Stoll: Theorem 6.3.2?

Stoll: Theorem 6.3.2 is a specific version of the Fundamental Theorem of Calculus. It states that if f(x) is a continuous function on the interval [a, b], then the integral of f(x) from a to b is equal to the limit of the sum of rectangles with base lengths equal to (b-a)/n and heights equal to f(x*) for some x* in the interval [a, b].

How is Stoll: Theorem 6.3.2 used in calculus?

Stoll: Theorem 6.3.2 is used in calculus to evaluate integrals, particularly those that cannot be evaluated using basic integration rules. It allows for the calculation of integrals by using the limit of a sum of rectangles, making it a powerful tool in solving complex integration problems.

What are the key assumptions of Stoll: Theorem 6.3.2?

The key assumptions of Stoll: Theorem 6.3.2 are that the function f(x) is continuous and the interval [a, b] is finite. This means that the function must be defined and have no discontinuities on the interval, and the interval cannot extend to infinity.

How does Stoll: Theorem 6.3.2 relate to the other versions of the Fundamental Theorem of Calculus?

Stoll: Theorem 6.3.2 is a specific version of the Fundamental Theorem of Calculus, which has three parts. Part 1 states that integration can be used to find the area under a curve, while part 2 states that the derivative of the integral of a function is the original function. Stoll: Theorem 6.3.2 is a more specific version of part 1, as it provides a method for calculating integrals using the limit of a sum of rectangles.

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