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Definition

Let $a, b \in R$
Suppose $a < b$
Suppose $f (x)$ is bounded on $[a, b]$
$\int_{a}^{b} f (x) d x$ exists and is equal if and only if there is a value that both the set of all lower darboux sums and upper darboux sums Converge to. $\int_{a}^{b} f (x) d x = s u p {L (f, p) ∣ P is any partiton of [a,b]} = in f {U (f, p) ∣ P is any partiton of [a,b]}$
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Text Elements

{ set of all L(f,P) } = { 1, 1.45, 1.495, 1.4995, … }

3

1

2

1

2

b

a

{ set of all U(f,P) } = { 2, 1.55, 1.505, 1.5005, … }
```
        When the two converge, meaning
```
inf( {set of all L(f,P)} ) = sup( {set of all U(f,P)} )

you get the integral. = 1.5
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