Second Mean Value Theorem for Integrals By Ng Tze Beng. The Second Mean Value Theorem for Integrals (SMVT) Statement of the Theorem

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1 Secod Me Vlue Theorem for Itegrls By Ng Tze Beg This rticle is out the Secod Me Vlue Theorem for Itegrls. This theorem, first proved y Hoso i its most geerlity d with extesio y ixo, is very useful d lmost idispesle i my of the rgumets i the covergece prolem of Fourier series. We shll preset proof log the lie tke y ixo d Hoso. The Secod Me Vlue Theorem for Itegrls (SMVT) Sttemet of the Theorem Suppose f is Leesgue itegrle o [, ] d g: [, ] R is mootoe. The (M) (i) f ( xgxdx ) ( ) g ( ) f( xdx ) g ( ) f( xdx ) for some with ; (ii) (M) holds with < < except i some trivil cses where g(x) is costt i the ope itervl (, ); (iii) (M) holds with g() d g() replced y A d B respectively so tht the fuctio A, x, hx ( ) gx ( ), x, B, x is mootoe; i.e. f ( xgxdx ) ( ) A f( xdx ) B f( xdx ) for some with ; A lim gx ( ), B lim gx ( ) if g is icresig d x x A lim gx ( ), B lim gx ( ) if g is decresig. c e tke i (, ) except i x x some trivil cses where g(x) is costt i the ope itervl (, ).

2 Note tht we sy fuctio g is decresig if x y g( x) g( y). It is sid to e icresig if x y g( x) g( y). A immedite cosequece is the followig: orollry (Boet Me Vlue Theorem). Suppose f is Leesgue itegrle o [, ] d g: [, ] R is mootoe. (i) If g is o-egtive, decresig d greter th or equl to 0, for A lim gx ( ) there exists such tht d x f ( xgxdx ) ( ) A f( xdx ). (ii) If g is o-egtive, icresig d greter th or equl to 0, for B lim gx ( ) there exists such tht d x f ( xgxdx ) ( ) B f( xdx ). (iii) If g is ot costt i (, ), the the poit i Prt (i) d (ii) is i (, ). More precisely, except for some trivil cses whe g is costt i (, ), the poit i Prt (i) d (ii) is i (, ). Proof. For Prt (i), pply SMVT Prt (iii) with B = 0. For Prt (ii), pply SMVT Prt (iii) with A = 0. Prt (iii) follows from SMVT Prt (iii). Note tht we revite the Secod Me Vlue Theorem for Itegrls y SMVT. efiitio. Let [, ] e closed d ouded itervl. A fiite fuctio g: [, ] R is sid to e step fuctio, if there exists prtitio of [, ], = x 0 < x < < x =, such tht g is costt o ech ope itervl ( xi, xi), i=, 2,,. We shll stte the followig pproximtio theorem of Leesgue itegrle fuctio without proof. This follows from the pproximtio of Leesgue itegrle fuctio y simple fuctio d the pproximtio of chrcteristic 2

3 fuctio of mesurle suset of fiite mesure y step fuctio. The proofs my e foud i Royde s Rel Alysis. Theorem 2. Suppose g is Leesgue itegrle o [, ]. For y > 0, there exists step fuctio o [, ] such tht g () t () t dt. Our pproch is to prove Prt (i) of SMVT for step fuctio d the use the pproximtio theorem to pss to geerl Leesgue itegrle fuctio. The first thig we oserve is tht Prt (i) of SMVT holds true whe f is costt fuctio. Lemm 3. Suppose f is costt fuctio o (, ) d g: [, ] R is decresig d g() > g(). The Prt (i) of SMVT holds true. Proof. Suppose f (x) is costt fuctio, g is decresig, g ( ) d g ( ) 0. Suppose f (x) = M i (, ). Sice = g() g(x) g() = 0, g( x) dx 0. Therefore, either M( ) f( x) g( x) dx M g( x) dx 0 or M( ) f( x) g( x) dx M g( x) dx 0. Hece for some 0 t, t( ) t( ) f ( xgxdx ) ( ) tm ( ) Mdt f( tdt ). Note tht t( ) d so we c tke to e equl to t( ) d we get t( ) f ( x) g( x) dx tm ( ) Mdt g( ) f ( t) dt. This proves the cse whe f is costt fuctio i (, ). 3

4 gx ( ) g ( ) I geerl, let gx ˆ( ) for x i [, ]. The ĝ is decresig, g ( ) g ( ) g ˆ( ) d g ˆ( ) 0. By wht we hve just proved, there exists such tht d f ( xgxdx ) ˆ( ) f( xdx ) () osequetly, f ( xgxdx ) ( ) g ( ) f( xdx ) ( g ( ) g ( )) f( xdx ) d Prt (i) of SMVT holds. The followig techicl lemm is used to comie the itervls o which Prt (i) of SMVT holds. Lemm 4. Assume tht g is decresig o [, ]. If Prt (i) of SMVT is true for the suitervls [, ] d [, ], the it is true for the itervl [, ]. Proof. As Prt (i) of SMVT is true for the itervl [, ], we hve f ( xgxdx ) ( ) g( ) f( xdx ) g( ) f( xdx ) (2) for some. Similrly, we get f ( xgxdx ) ( ) g( ) f( xdx ) g( ) f( xdx ) (3) for some. We c write (2) d (3) s follows: f ( xgxdx ) ( ) g( ) f( xdx ) g( ) f( xdx ) f( xdx ) (4) f ( xgxdx ) ( ) g( ) f( xdx ) f( xdx ) g( ) f( xdx ) f( xdx ) (5) Thus ddig (4) d (5) gives 4

5 f ( xgxdx ) ( ) g( ) g( ) f( x) dx g( ) g( ) f( x) dx g( ) f( x) dx (6). If g( ) g( ), the g is costt o [, ]. The from (6) we oti f ( xgxdx ) ( ) g( ) f( xdx ) g( ) f( xdx ) g( ) f( xdx ) for y with < <. Now ssume g( ) g( ). Oserve tht g( ) g( ) g( ) g( ) f ( xdx ) f( x) g( ) g( ) g( ) g( ) is of the form f ( xdx ) m f( x) with, m 0 d m. x Sice the fuctio H ( x) f ( t) dt is cotiuous, there exists E with E such tht Thus g( ) g( ) g( ) g( ) E f ( xdx ) f( x) f( x) g( ) g( ) g( ) g( ). E. g( ) g( ) f( xdx ) g( ) g( ) f( xdx ) g( ) g( ) f( xdx ) It follows the from (6), f ( xgxdx ) ( ) g( ) g( ) f( xdx ) g( ) f( xdx ) E g ( ) f ( x ) dx g ( ) f ( x ) dx E. Thus Prt (i) of SMVT holds o the itervl [, ]. This completes the proof of Lemm 4. E 5

6 Proof of the Secod Me Vlue Theorem Prt (i). We prove Prt (i) of SMVT whe g is decresig. The cse whe g is icresig follows y cosiderig g, sice g is decresig. If g() = g(), the g is costt, sy g(x) = K, the we c tke to e y vlue i (, ), sice trivilly, f ( x) g( x) dx Kf( x) dx g( ) f( x) dx g( ) f( x) dx. We ssume ow g() > g(). Let gx ˆ( ) g ˆ( ) 0. gx ( ) g ( ) g ( ) g ( ) for x i [, ]. The ĝ is decresig, g ˆ( ) d x Let F( x) f( t) dt for x i [, ]. The F is cotiuous o [, ]. It is sufficiet to show tht there exists such tht d This is ecuse if (7) holds, the Ad we hve d so f ( xgxdx ) ˆ( ) F ( ) (7) g( x) g( ) f ( xgxdx ) ˆ( ) f( x) dxf ( ) f( xdx ) g ( ) g ( ). f ( xgxdx ) ( ) g ( ) f( xdx ) g ( ) g ( ) f( xdx ) f ( xgxdx ) ( ) g ( ) f( xdx ) g ( ) f( xdx ) f( xdx ) g( ) f( x) dx g( ) f ( x) dx Thus Prt (i) of SMVT follows.. 6

7 Note tht with g ˆ( ) d g ˆ( ) 0, (7), i.e., f ( xgxdx ) ˆ( ) F ( ) f( xdx ) g ˆ( ) f( xdx ) g ˆ( ) f( xdx ) is specil cse of the theorem. se. f is costt fuctio o (, ). By Lemm 3, whe f (x) is costt fuctio o (, ), Prt (i) of SMVT holds. se 2. f is step fuctio, g is decresig. Sice f is step fuctio, there is prtitio of [, ], = x 0 < x < < x =, such tht f is costt o ech ope itervl ( xi, xi), i=, 2,,. By se, we kow tht Prt (i) of SMVT is true o ech suitervl [ xi, xi] i=, 2,,. Sice there is oly fiite umer of suitervls i the prtitio, y Lemm 4, strtig from the first itervl, we c exted to the whole itervl [, ] for Prt (i) of SMVT to hold. Thus Prt (i) of SMVT is true for step fuctios. se 3. Geerl f., Suppose the fuctio f is Leesgue itegrle o [, ]. The y Theorem 2, for y iteger, there exists step fuctio such tht () t f() t dt (8) Therefore, for ll x i [, ], x x x () tdt f() tdt () t f() t dt. Hece, x x x Fx ( ) f() tdt () tdt f() tdt Fx ( ) (9) Sice F is cotiuous o [, ], y the Extreme Vlue Theorem, there exists, i [, ], such tht F( ) if{ F( x): x[, ]} m d F( ) sup{ F( x): x[, ]} M. It follows the from (9) tht 7

8 x m () t dt M (0) Sice Prt (i) of SMVT holds for step fuctio, for ech, there exists such tht d ( ) ˆ x g( x) dx ( x) dx () Therefore, it follows from (0) tht for, Now Therefore, m ( ) ˆ x g( x) dx M (2) gxf ˆ( ) ( xdx ) gx ˆ( ) ( xdx ) gx ˆ( ) f( x) ( x) dx f( x) ( x) dx y (8). gx ˆ( ) ( ) ˆ( ) ( ) ˆ xdx gxf xdx gx ( ) ( xdx ) (3) We deduce from (2) d (3) tht 2 2 m gˆ( x) f( x) dx M (4) Therefore, lettig teds to, we hve m gˆ( x) f( x) dx M d so y the Itermedite Vlue Theorem, there exists lyig etwee d d hece i [, ] such tht F ( ) gx ˆ( ) f( xdx ). Thus we hve proved (7) d Prt (i) of SMVT follows. 8

9 Proof of Prt (ii) of the Secod Me Vlue Theorem. We prove Prt (ii) of SMVT whe g is decresig. The cse whe g is icresig follows y cosiderig g. By prt (i), gxf ˆ( ) ( xdx ) F ( ) for some i [, ]. If lies i (, ), the prt (ii) follows. If F() F() = 0, the gxf ˆ( ) ( xdx ) 0, i.e., g( x) f( x) dx g( ) f( x) dx., i.e., If F() F(), the gˆ( x) f( x) dx f( x) dx g( x) f( x) dx g( ) f( x) dx( g( ) g( )) f( x) dx g( ) f( x) dx. Therefore, Prt (ii) of SMVT will hold uless F() = F() = 0 or F() = F() d for ll x i (, ), F( x) F( ) d F( x) F( ). We shll show tht uder this coditio g must e costt i (, ). The coditio implies tht F( x) F( ) for ll x i (, ). It follows the y cotiuity tht G(x) = F(x) F() is of costt sig. Note tht F() = 0 or F(). Our pproch will e to show tht for y ope itervl (c, d) i [, ], g is costt. Let < c < d <. We pply Prt (i) of SMVT to ech of the three itervls, [, c], [c, d] d [d, ]. We oti c c f ( xgxdx ) ˆ( ) g ˆ( ) ( ) ˆ( ) ( ) f xdxgc f xdx, d d 2 f ( xgxdx ) ˆ( ) gc ˆ( ) ( ) ˆ( ) ( ) c f xdxgd f xdx c d 2 3 f ( xgxdx ) ˆ( ) gd ˆ( ) ( ) ˆ( ) ( ) d f xdxg f xdx d (5) 3 Now writig (5) i terms of F(x) d otig tht g ˆ( ) d g ˆ( ) 0, we get: 9

10 c f ( xgxdx ) ˆ( ) F ( ˆ ) gc ( ) Fc ( ) F ( ), c d f ( xgxdx ) ˆ( ) gc ˆ( ) F ( ) Fc ( ) gd ˆ( ) Fd ( ) F ( ) 2 2 d f ( xgxdx ) ˆ( ) gd ˆ( ) F ( 3) Fd ( ), (6) d for some c2 d 3 Addig ll three equtios i (6) we oti f ( xgxdx ) ˆ( ) F ( ) gc ˆ( ) F ( ) gc ˆ( ) gd ˆ( ) F ( ) gd ˆ( ) (7) 2 3 Sice gxf ˆ( ) ( xdx ) F ( ), we get from (7) tht F( ) F( g( c) ˆ F ( ) F ( ) gc ˆ( ) gd ˆ( ) F ( ) F ( ) gd ˆ( ) (8) 2 3 Note tht sice gxis ˆ( ) decresig o [, ], gc ˆ( ) 0, gc ˆ( ) gd ˆ( ) 0, gd ˆ( ) g ˆ( ) 0. If G(x) = F(x) F() > 0 i (, ), the sice F is cotiuous o [, ], G(x) 0 o [, ]. If G(x) = F(x) F() < 0 i (, ), the G(x) 0 o [, ]. Therefore, ech of the three terms i (8) is of the sme sig or equl to 0. Hece ech of the three terms i (8) must vish. osequetly, sice F ( ) F ( ) 0 for 2 is i (, ), gc ˆ( ) gd ˆ( ) 0. This implies tht 2 gc ( ) gd ( ). Therefore g must e costt o [c, d]. Sice [c, d] is y itervl i (, ), this implies tht g is costt i (, ). This proves Prt (ii) of SMVT for decresig g. Proof of Prt (iii) Secod Me Vlue Theorem. Suppose f is Leesgue itegrle o [, ] d g: [, ] R is mootoe. 0

11 A, x, osider the fuctio hx ( ) gx ( ), x, which is mootoe o [, ]. B, x This mes A lim gx ( ), B lim gx ( ) if g is icresig d x x A lim gx ( ), B lim gx ( ) if g is decresig. x x Therefore, pplyig Prt (i) of SMVT to h we get f ( xgxdx ) ( ) f( xhxdx ) ( ) h ( ) f( xdx ) h ( ) f( xdx ), A f( x) dxb f( x) dx for some with. If g is ot costt o (, ), the the poit c e tke to e i (, ). More precisely, y Prt (ii) of SMVT, c e tke to e i (, ) except for some trivil cses whe g is costt i (, ). This completes the proof of the Secod Me Vlue Theorem for Itegrls.

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right: Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the