Darboux & ε-Reformulation of the Definite Integral

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11/7/2020

Darboux Definition of the Definite Integral

ε-Reformulation Definition of the Definite Integral

HOW and WHEN to use these definitions to:

1. Prove when a function is integrable over a specified interval [a, b]

2. Prove the definite integral properties (Theorem 4 through Theorem 7 in Spivak)

But first...

Darboux definition of the Definite Integral

Let \space a, b \in \R, a < b.

Suppose \space f \space is\space bounded.

Let \space P = \{ti\}, where \space 0\leq i \leq n.

Then \space we \space say \space f \space is \space integrable \space on \space [a,b] \space IFF

sup\{L(f, P)\} = inf\{U(f, P)\} = Area \space

What does this mean geometrically speaking?

Darboux definition of the Definite Integral

Lower sum of f on P

Higher sum of f on P

ε-Reformulation definition of the Definite Integral (restatement)

If \space f \space is \space bounded \space on \space [a, b] \space then \space f \space is \space integrable

on \space [a, b] \space IFF \space \forall \epsilon > 0, \space there \space is \space a \space partition \space P

of \space [a, b] \space s.t, \space U(f, P) - L(f, P) < \epsilon

What does this mean geometrically speaking? How is this a restatement?

ε-Reformulation definition of the Definite Integral (restatement)

If we choose a partition and compute its upper and lower sums and the difference between these sums is < ε, then the upper and lower sum estimates can be made infinitely close to one another. I.e, they result in practically the same number and represent the area!

Recall ε is an infinitely small number

restatement bc this is the same idea as Darboux: upper and lower sums equal to each other and the area of the curve

When/how do we use these definitions when we have the Riemann sum definition?

WHEN: to prove that a function, f, is not integrable over an interval [a, b].

HOW:
Darboux defn:

We simply show that the supremum of the lower sums of f on P does not equal to the infimum of the upper sums f on p.

Epsilon defn:

We assume that the definition is false, and choose a value of epsilon (after computing U(f, P) - L(f, P) that contradicts the definition.

When/how do we use these definitions when we have the Riemann sum definition?

WHEN: to prove that a function, f, is integrable over an interval [a, b].

HOW:
Darboux defn:

We prove the definition. I.e, compute the lower and upper sums of f on P and show the infimum of the upper sum and supremum of the lower sum are equivalent.

Epsilon defn:

We choose a value of epsilon that is greater than the difference of the upper and lower sum of f on P

Prove a fcn is integrable over [a, b]