Cii integral 5
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The document discusses Riemann integration and Darboux integration. It defines the Riemann sum of a function f over a partition P with tags T. It states that a necessary and sufficient condition for a function to be Riemann integrable is for the Riemann sums to converge to the same limit regardless of the partitions and tags chosen. It also shows that if a function is Darboux integrable, then it is also Riemann integrable and the integrals are equal.
![Pr. 8
f (x) monìtonh k—i fr—gmènh sto [a, b] f (x) @DarbouxAEoloklhr¸simh
Apìdeixh.
istw f (x) —Ôxous— k—i P = {x0 , x1 , . . . , xn } miˆ di—mèrish k—i
xk ≤ |P|
en xk−1 ≤ x ≤ xk tìte mk = inf {f (x), x ∈ [xk−1 , xk ]} = f (xk−1 ) k—i
Mk = sup {f (x), x ∈ [xk−1 , xk ]} = f (xk ) opìte
n
U(P, f ) − L(P, f ) = (Mk − mk ) xk ≤
k=1
n
≤ |P| (Mk − mk ) = |P| (f (b) − f (a))
k=1
qi— kˆje > 0 upˆrqei mi— di—mèrish me |P| < opìte
f (b) − f (a)
U(P, f ) − L(P, f ) < F epì thn €rF T sunepˆget—i ìti upˆrqei to
olokl rwm— DarbouxF
Pr. 9
f (x) suneq ™ sto [a, b] f (x) omoiìmorf— suneq ™ sto [a, b]
f (x) @DarbouxAEoloklhr¸simh
Apìdeixh.
f (x) omoiìmorf— suneq ™ ⇔
∀ > 0, ∃ δ( ) : |x − y | < δ( ) |f (x) − f (y )| <
b−a
hi—lègoume mi— di—mèrish me |P| < δ( )F ipeid h sunˆrthsh eÐn—i
@omoiìmorf—A suneq ™ sto diˆsthm— [xk−1 , xk ] j— èqoume
Mk = sup {f (x), x ∈ [xk−1 , xk ]} = f (ξk )
mk = inf {f (x), x ∈ [xk−1 , xk ]} = f (ηk )
me ξk , ηk ∈ [xk−1 , xk ] opìte Mk − mk = f (ξk ) − f (ηk ) < k—i
b−a
n
U(P, f ) − L(P, f ) = (Mk − mk ) xk <
k=1
n
≤ xk =
b−a
k=1
epì thn €rF T sunepˆget—i ìti upˆrqei to olokl rwm— DarbouxF](https://image.slidesharecdn.com/ciiintegral5-120718090455-phpapp02/85/Cii-integral-5-8-320.jpg)
![Pr. 8
f (x) monìtonh k—i fr—gmènh sto [a, b] f (x) @DarbouxAEoloklhr¸simh
Pr. 9
f (x) suneq ™ sto [a, b] f (x) omoiìmorf— suneq ™ sto [a, b]
f (x) @DarbouxAEoloklhr¸simh](https://image.slidesharecdn.com/ciiintegral5-120718090455-phpapp02/85/Cii-integral-5-9-320.jpg)
![Pr. 12 Idiìthtec oloklhrwmˆtwn
Grammikìthta
b b b
(c1 f (x) + c2 g (x)) dx = c1 f (x) dx + c2 g (x) dx
a a a
"Jetikìthta
b b
f1 (x) ≤ f2 (x) f1 (x) dx ≤ f2 (x) dx
a a
Trigwnik idiìthta
b b
f (x) dx ≤ |f (x)| dx
a a
Qwrismìc diast matoc
b c b
c ∈ [a, b] f (x) dx = f (x) dx + f (x) dx
a a c
Epektˆseic oloklhr¸matoc
a b a
or or
f (x) dx ≡ − f (x) dx f (x) dx ≡ 0
b a a
b
m ≤ f (x) ≤ M m(b − a) ≤ f (x) dx ≤ M(b − a)
a](https://image.slidesharecdn.com/ciiintegral5-120718090455-phpapp02/85/Cii-integral-5-13-320.jpg)
Cii integral 5
- 1. Darboux olokl rwma Kˆtw olokl rwma L(f ) ≡ sup L(P, f ) P
- 2. Anw olokl rwma U(f ) ≡ inf U(P, f ) P
- 3. Olokl rwma Darboux f (x) Darboux oloklhr¸simh ⇔ L(f ) = U(f ) b L(f ) = U(f ) ≡ ID (f ) ≡ f (x) dx a uˆtw olokl rwm— enw olokl rwm—
- 4. Pr. 5 en upˆrqei Pn eÐn—i kˆpoi— —koloujЗ di—merÐsewn tètoi— ¸ste lim L(Pn , f ) = lim U(Pn , f ) n→∞ n→∞ tìte h sunˆrthsh f (x) eÐn—i oloklhr¸simh L mma ∀ > 0 ∃ P : U(P, f ) − L(P, f ) < U(f ) − L(f ) + Pr.6 f (x) eÐn—i oloklhr¸simh L(f ) = U(f ) ⇔ ∀ > 0 ∃ P : U(P, f ) − L(P, f ) < Pr.7 Je¸rhma Darboux f (x) eÐn—i oloklhr¸simh L(f ) = U(f ) ⇔ ∀ > 0 ∃ δ > 0 : |P| < δ ⇒ U(P, f ) − L(P, f ) <
- 5. Pr. 5 en upˆrqei Pn eÐn—i kˆpoi— —koloujЗ di—merÐsewn tètoi— ¸ste lim L(Pn , f ) = lim U(Pn , f ) n→∞ n→∞ tìte h sunˆrthsh f (x) eÐn—i oloklhr¸simh Apìdeixh. iqoume ìti L(Pn , f ) ≤ L(f ) ≤ U(f ) ≤ U(Pn , f ) ⇒ lim L(Pn , f ) ≤ L(f ) ≤ U(f ) ≤ lim U(Pn , f ) n→∞ n→∞ L mma ∀ > 0 ∃ P : U(P, f ) − L(P, f ) < U(f ) − L(f ) + Apìdeixh. ipeid L(f ) = sup L(P, f ) ⇒ ∀ > 0 ∃ P1 : L(P1 , f ) > L(f ) − /2 P U(f ) = inf U(P, f ) ⇒ ∀ > 0 ∃ P2 : U(P2 , f ) < U(f ) + /2 P k—i gi— P = P1 P2 L(P, f ) ≥ L(P1 , f ) > L(f ) − /2 U(P, f ) ≤ U(P2 , f ) > U(f ) + /2 ˆr— U(P, f ) − L(P, f ) < U(f ) − L(f ) +
- 6. Pr.6 f (x) eÐn—i oloklhr¸simh L(f ) = U(f ) ⇔ ∀ > 0 ∃ P : U(P, f ) − L(P, f ) < Apìdeixh. en L(f ) = U(f ) tìte —pì to p—r—pˆnw l mm— èqoume ìtiX ∀ > 0 ∃ P : U(P, f ) − L(P, f ) < istw t¸r— ìti h p—r—pˆnw sqèsh —lhjeÔei tìte èqoume epÐsh™ ìtiX L(P, f ) ≤ L(f ) ≤ U(f ) ≤ U(P, f ) ⇒ U(f )−L(F ) ≤ U(P, f )−L(P, f ) < er— ∀ > 0 U(f ) − L(F ) < opìte L(f ) = U(f )F
- 7. Pr.7 Je¸rhma Darboux f (x) eÐn—i oloklhr¸simh L(f ) = U(f ) ⇔ ∀ > 0 ∃ δ > 0 : |P| < δ ⇒ U(P, f ) − L(P, f ) < Apìdeixh. en ∀ > 0 ∃ δ > 0 : |P| < δ ⇒ U(P, f ) − L(P, f ) < tìte ik—nopoioÔnt—i oi sunj ke™ th™ prìt—sh™ TD epeid ∀ > 0 ∃ P : U(P, f ) − L(P, f ) < ⇒ L(f ) = U(f ) istw t¸r— L(f ) = U(f ) tìte sÔmfwn— me thn prìt—sh T ∀ > 0 ∃ P0 : U(P0 , f ) − L(P0 , f ) < /2 epì thn di—mèrish P0 orÐzoume δ= |f (x)| < B 8d(P0 )B istw mi— di—mèrish P, |P| < δ F yrÐzoume Q = P P0 d(Q) − d(P) ≤ d(P0 ) epì thn prìt—sh P L(Q, f )−L(P, f ) ≤ 2 (d(Q) − d(P)) B|P| ≤ 2d(P0 )B|P| < 2d(P0 )Bδ = 4 —pì thn prìt—sh P èqoume L(P0 , f ) − L(P, f ) ≤ L(Q, f ) − L(P, f ) < 4 ìmoi— —podeiknÔoume ìtiX U(P, f ) − U(P0 , f ) < 4 epomènw™ U(P, f ) − L(P, f ) = = U(P, f ) − U(P0 , f ) + U(P0 , f ) − L(P0 , f ) + L(P0 , f ) − L(P, f ) < < /4 < /2 < /4
- 8. Pr. 8 f (x) monìtonh k—i fr—gmènh sto [a, b] f (x) @DarbouxAEoloklhr¸simh Apìdeixh. istw f (x) —Ôxous— k—i P = {x0 , x1 , . . . , xn } miˆ di—mèrish k—i xk ≤ |P| en xk−1 ≤ x ≤ xk tìte mk = inf {f (x), x ∈ [xk−1 , xk ]} = f (xk−1 ) k—i Mk = sup {f (x), x ∈ [xk−1 , xk ]} = f (xk ) opìte n U(P, f ) − L(P, f ) = (Mk − mk ) xk ≤ k=1 n ≤ |P| (Mk − mk ) = |P| (f (b) − f (a)) k=1 qi— kˆje > 0 upˆrqei mi— di—mèrish me |P| < opìte f (b) − f (a) U(P, f ) − L(P, f ) < F epì thn €rF T sunepˆget—i ìti upˆrqei to olokl rwm— DarbouxF Pr. 9 f (x) suneq ™ sto [a, b] f (x) omoiìmorf— suneq ™ sto [a, b] f (x) @DarbouxAEoloklhr¸simh Apìdeixh. f (x) omoiìmorf— suneq ™ ⇔ ∀ > 0, ∃ δ( ) : |x − y | < δ( ) |f (x) − f (y )| < b−a hi—lègoume mi— di—mèrish me |P| < δ( )F ipeid h sunˆrthsh eÐn—i @omoiìmorf—A suneq ™ sto diˆsthm— [xk−1 , xk ] j— èqoume Mk = sup {f (x), x ∈ [xk−1 , xk ]} = f (ξk ) mk = inf {f (x), x ∈ [xk−1 , xk ]} = f (ηk ) me ξk , ηk ∈ [xk−1 , xk ] opìte Mk − mk = f (ξk ) − f (ηk ) < k—i b−a n U(P, f ) − L(P, f ) = (Mk − mk ) xk < k=1 n ≤ xk = b−a k=1 epì thn €rF T sunepˆget—i ìti upˆrqei to olokl rwm— DarbouxF
- 9. Pr. 8 f (x) monìtonh k—i fr—gmènh sto [a, b] f (x) @DarbouxAEoloklhr¸simh Pr. 9 f (x) suneq ™ sto [a, b] f (x) omoiìmorf— suneq ™ sto [a, b] f (x) @DarbouxAEoloklhr¸simh
- 10. Olokl rwma Riemann diamèrish P = {x0 , x1 , x2 , . . . , xn , } epilog shmeÐwn T = {ξ1 , ξ2 , . . . , ξn , } ìpou xk−1 ≤ ξk ≤ xk ˆjroisma Riemann n S (P, T , f ) = f (ξk ) xk k=1
- 11. Orismìc: Olokl rwma Riemann ∃ olokl rwm— ⇔ ∃ lim S (P, T , f ) = IR (f ) Riemann |P|→0 ∀ > 0 ∃ δ( ) > O : ∀ |P| < δ k—i T epilog shmeÐwn ⇒ |S (P, T , f ) − IR (f )| < en upˆrqei to olokl rwm— Riemann tìte eÐn—i mon—dikì
- 12. Pr. 10 olokl rwm— olokl rwm— X IR (f ) = ID (f ) Riemann Darboux Pr. 11 (Je¸rhma Darboux) k −1 k ξk,n ∈ a + (b − a), a + (b − a) n n b n b−a f (x) dx = lim f (ξk,n ) n→∞ n a k=1 €—rˆdeigm— n n 1 1 1 1 1 lim = lim = dx = ln 2 n→∞ n+k n→∞ n k 1+x k=1 k=1 n + 0 n
- 13. Pr. 12 Idiìthtec oloklhrwmˆtwn Grammikìthta b b b (c1 f (x) + c2 g (x)) dx = c1 f (x) dx + c2 g (x) dx a a a "Jetikìthta b b f1 (x) ≤ f2 (x) f1 (x) dx ≤ f2 (x) dx a a Trigwnik idiìthta b b f (x) dx ≤ |f (x)| dx a a Qwrismìc diast matoc b c b c ∈ [a, b] f (x) dx = f (x) dx + f (x) dx a a c Epektˆseic oloklhr¸matoc a b a or or f (x) dx ≡ − f (x) dx f (x) dx ≡ 0 b a a b m ≤ f (x) ≤ M m(b − a) ≤ f (x) dx ≤ M(b − a) a