BULETNUL Unverstăţ Petrol Gae dn Ploeşt Vol. LV No. /6 7 - Sera Mateatcă - noratcă - Fcă The Aroate Calculaton o Dente Sle and Multle nterals Tănase Dnu Unverstatea Petrol-Gae dn Ploeşt Bd. Bucureşt 9 Ploeşt Catedra de Mateatcă e-al: tdnu@al.u-loest.ro Abstract Ths aer resents the local scheata and the roras n turbo-ascal or the aroate calculaton ethods o the sle dente nteral [] o the double dente nteral [] and o the trle dente nteral []. At the end there are analed the calculaton errors corresondn to these ethods that use the olnoals o nterolaton to ve nodes. Ke words: dvson nterolaton alorth ntroducton Paer [] resents ethod B or the aroate calculaton o sle dente nteral where :[ a b] R s a nterable uncton on [ ] t s consdered an equdstant dvson wth Δ : wth the dstance o the dvson densons: b ( ) d () a a b. n nodes. ( a < <... < < < < b)... n b a h then the aroate value o the nteral n n n h 7( ( a) ( b) ) ( ) ( ) 5 n n ( ) ( ) () or whch the calculaton error s: where: e M 5 ( b a) 6 τ h ()
8 Tănase Dnu su ( 5) ( ) 5 M cu C [ ab ]. [ a b] n aer [] the BB ethod descrbes the aroate calculaton o the double dente nteral: wth : D R nterable where a b. wth ( ) ( ) or an [ ] D ( ) dd () D R and t s bordered b both ( ) The aroate value o the nteral s ven b the orula: n n n h 7 n 5 where: ( ϕ ( ) ϕ( )) ϕ( ) ϕ( ) ϕ( ) wth ( ) n ( ) ϕ (5) h ϕ( ) ( ( ) ( )) ( ) ( ) 7 5 ( ) ( ) (6) b a where the nor o the nterval [a b] s: h and the nor o the nterval [ ( ) ( )] n ( ) ( ) s: h wth {... n}. Paer [] resents the ethod BBB or the aroate calculaton o the trle dente nterals. wth : V R nterable on V ( ) ddd (7) V R. The doan V s dened b suraces: ( ) and ( ) wth ( ) ( ) or ( )( ) D where D s the roecton o the doan V on the O lan and t s dened b the curves: ( ) and ( ) wth ( ) ( ) or ( ) [ a b] where [a b] s the roecton o D on the as O. The aroate value o the nteral s ven b the orula: n n ( ( ) ( )) ( ) ( ) ( ) n h 7 ϕ ϕ n ϕ ϕ ϕ 5 n ( ) ϕ (8) b a h ν Δ and n where the nterval [a b] has the nor ( )
The Aroate Calculaton o Dente Sle and Multle nterals 9 ( ) ( ) ( ) ( ) ( ) ( ) 7 5 h ϕ ( ) ( ) (9) where the nterval ( ) ( ) [ ] has the nor ( ) ( ) ( ) h Δ ν and ( ) ( ) ( ) ( ) ( ) 7 5 h ( ) ( ) ( ) () or { } { } n...... and the nterval ( ) ( ) [ ] has the nora o the equdstant dvson ( ) ( ) ( ) h Δ ν. Contents A. Proras rora sle_nteral; te vecarra[..5] o real; var a:vec; absssht:real; n:nteer; uncton su(var a:vec; :nteer; h:real):real; var :nteer; ssssss:real; ben s:; s:; s:; s:; or : to - do ben s:sa[*]; s:sa[*]; s:sa[*]; s:sa[*] s:s-a[]; ss:7*(a[] a[*n]); ss:ss*s*s*s*s; ss:(ss**h)/5; su:ss
Tănase Dnu uncton un(var :nteer; ah:real):real; var :real; ben :a*h; un:(*)/(**) ben {clrscr;} wrte('a'); readln(a); wrte('b'); readln(b); wrte('n'); readln(n); h:(b-a)/(*n); or : to *n do a[]:un(ah); s:su(anh); wrteln('nteral 's::6); {reeat untl eressed} end. rora double_nteral; te vecarra[..5] o real; var c:vec; absshabh:real; n:nteer; uncton su(var a:vec;:nteer;h:real):real; var :nteer; ssssss:real; ben s:; s:; s:; s:; or : to - do ben s:sa[*]; s:sa[*]; s:sa[*]; s:sa[*] s:s-a[]; ss:7*(a[] a[*n]); ss:ss*s*s*s*s; ss:(ss**h)/5; su:ss uncton un(:nteer; ahah:real):real; var :real; ben :a*h; :a*h; un:/sqrt(**)
The Aroate Calculaton o Dente Sle and Multle nterals uncton (var :nteer; ah:real):real; var t:real; ben :a*h; t:sqrt(abs(-*)); :t; uncton (var :nteer; ah:real):real; var t:real; ben :a*h; t:-sqrt(abs(-*)); :t ben {clrscr; } wrte('a'); readln(a); wrte('b'); readln(b); wrte('n'); readln(n); wrte(''); readln(); h:(b-a)/(*n); or : to *n do ben a:(ah); b:(ah); h:(b-a)/(*); or : to * do c[]:un(ahah); []:su(ch); ss:su(nh); wrteln('nteral ' ss::6) end. rora treble_nteral; {uses crt;} te vecarra[..5] o real; var c:vec; abhabhabhss:real; n:nteer; uncton su(var a:vec;:nteer;h:real):real; var :nteer; ssssss:real; ben s:; s:; s:; s:; or : to - do ben s:sa[*]; s:sa[*];
Tănase Dnu s:sa[*]; s:sa[*] s:s-a[]; ss:7*(a[] a[*n]); ss:ss*s*s*s*s; ss:(ss**h)/5; su:ss; uncton un(var :nteer; ahahah:real):real; var t:real; ben :a*h; :a*h; :a*h; t:sqrt(***); un:t uncton (var :nteer; ab:real):real; var :real; ben :a*b; :sqrt(abs(9-*)) uncton (var :nteer; ab:real):real; var :real; ben :a*b; :-sqrt(abs(9-*)) uncton (var :nteer; ahah:real):real; var t:real; ben :a*h; :a*h; t:; :t uncton (var :nteer; ahah:real):real; var :real; ben :a*h; :a*h; :(**)/9 ben {clrscr;} wrte('a'); readln(a); wrte('b'); readln(b); wrte('n'); readln(n);
The Aroate Calculaton o Dente Sle and Multle nterals wrte(''); readln(); wrte(''); readln(); h:(b-a)/(*n); or : to *n do ben a:(ah); b:(ah); h:(b-a)/(*); or : to * do ben a:(abab); b:(abab); h:(b-a)/(*); or : to * do c[]:un(ahahah); []:su(ch); []:su(h); ss:su(nh); wrteln('nteral ' ss::6) end. B. Alcatons Let us calculate the ollown dente nterals: D ( ) dd ; d where D ( ) { 9}; ddd where V ( ) 9 and [ ] V usn the anteror roras. Soluton Drectl calculated b eans o the analtcal ethods we obtan: ln 96; 58599 { } 988 Aroatel calculated b eans o B BB BBB ethods and usn the roraes resented above we obtan: 9 585985 988 or n a b ; or n a b ; or n a b.
Tănase Dnu Reerences. Dnu T. - Anală nuercă Edtura U.P.G. dn Ploeşt. D n u T. - Calculul aroatv al nteralelor sle Buletnul U.P.G. dn Ploeşt nr./. D n u T. - Calculul aroatv al nteralelor duble Buletnul U.P.G. dn Ploeşt nr./5. D n u T. - The Rouh Calculus o Treble Dened nteral Buletnul U.P.G. dn Ploeşt nr./6. -6 5. Grore Gh. - Lecţ de Anală nuercă Facultatea de Mateatcă Bucureşt (curs ltoraat) 6. Marnescu Gh. - Anală nuercă Edtura Acadee Roâne Bucureşt 97 Calculul aroatv al nteralelor dente sle ş ultle Reuat În această lucrare sunt reentate scheele loce ş roraele în Turbo-Pascal entru etodele de calcul aroatv ale nteralelor dente sle [] ale nteralelor dente duble [] ş ale nteralelor dente trle []. În nal sunt analate erorle de calcul coresunătoare acestor etode care olosesc olnoaele de nterolare cu cnc nodur