7.2 The Definition of the Riemann Integral. Outline

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1 7.2 The Definition of the Riemnn Integrl Tom Lewis Fll Semester 2014 Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Outline Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility

2 Bernhrd Riemnn ( ) Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Gston Drboux ( )

3 Definition Let J be ny bounded intervl of rel numbers, nd let f be bounded (rel-vlued) function on J. We define 1. M(f ; J) = sup x J f (x), 2. m(f ; J) = inf x J f (x), 3. ω(f ; J) = M(f, J) m(f, J). Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Definition A subdivision or prtition of the intervl [, b] is finite set {x 0, x 1,..., x n } such tht The induced subintervls = x 0 < x 1 < x 2 < < x n = b. I 1 = [x 0, x 1 ], I 2 = [x 1, x 2 ],... I n = [x n 1, x n ] re clled the components of σ. Given two prtitions σ nd τ of [, b], we sy tht τ is refinement of σ provided tht σ τ.

4 Definition (Upper nd lower sums) We cll n U(f ; σ) = M(f ; I k ) I k k=1 the upper sum for f nd σ. Likewise we cll L(f ; σ) = n m(f ; I k ) I k k=1 the lower sum for f nd σ. Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Lemm Let f be bounded function on [, b] nd let σ nd τ be prtitions of [, b] with σ τ. Then U(f ; σ) U(f ; τ) nd L(f ; σ) L(f ; τ).

5 Theorem Let f be bounded function on [, b]. Then every upper sum for f is greter thn or equl to every lower sum for f. Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Corollry Let f be bounded function on [, b]. Then inf U(f ; σ) sup L(f ; σ), where the infimum nd supremum re tken over ll prtitions of [, b].

6 Definition Let f be bounded function on [, b]. The upper integrl of f on [, b] is given by f (x)dx = f = inf U(f ; σ). The lower integrl of f on [, b] is given by f (x)dx = f = sup L(f ; σ). Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Note Recll tht inf U(f ; σ) sup L(f ; σ) from which it follows tht f f.

7 Definition Let f be bounded function on [, b]. We sy tht f is Riemnn integrble on [, b], written f R[, b], provided tht f = f. Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Problem For x [0, 1], let f (x) = { 1 if x is rtionl 0 if x is irrtionl. Evlute 1 0 f nd 1 0 f.

8 Theorem Let f be bounded function on [, b]. Then f R[, b] if nd only if for every ε > 0 there exists prtition σ of [, b] such tht U(f ; σ) L(f ; σ) < ε. Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Theorem If f is bounded nd monotone incresing (decresing) on [, b], then f R[, b].

9 Theorem (Exercise 6 from 7.2) If f is continuous on [, b], then f R[, b].