3 NTEGRATON tgrtio is us to swr qustios rltig to Ar Volum Totl qutity such s: Wht is th wig r of Boig 747? How much will this yr projct cost? How much wtr os this rsrvoir hol? How much ir is rquir y th popl i this lctur thtr urig this lctur? t is s o th cocpt of summtio of ifiitsimls. this lctur w will look t th o vril cs oly. Nt trm w will look t multipl itgrtio.
3. Dfiit tgrls Oft cll Rim itgrls ftr Brhr Rim (85-866). Th fiit itgrl of fuctio f () from to is fi s: y y f() f ( ) Ar ou y th curv y f ( ) lis y, (, r limits) y Rmmr tht rs low th is r gtiv v -v
This r c clcult y iviig it ito strips of with δ y y f() δ Cosir th r of th strip tw r A pproimtio to th r is δa f ( f ( ). r ( ) ) r r r r δ r r δ. ζ r r r Thr will som vlu of This pproimtio will gt ttr s δ. ζ such tht δ A f ( ζ ) δ r r r Th th r S r f ( ζ ) δ ctly s δ, ζ r r, w fi r f ( ) lim δ r f ( ζ ) δ r 3
Cosir th r, A( ) f ( ) f ( t) t i which t is ummy vril of itgrtio. (w us t to voi cofusio with, th uppr limit i this cs) y f(ζ) Cosir δ A( δ) f ( t) t A δa A() δa Agi, thr is som ζ such tht < ζ < δ, for which δ A( ) f ( ζ ) δ (s δ, f ( ζ ) f ( ) ). δ t() By fiitio, ζ A( ) lim A( δ) A( ) lim f ( ζ ) δ f ( ) δ δ δ δ i.. tgrtio is th rvrs procss to Diffrtitio 4
Cosir y fuctio F ( ) A( ) C itgrt tw,. Th F( ) A( ),, sic A ( ), C F() Th f ( t)t F( ) F( ) For, th fiit itgrl is f ( t)t F( ) F( ). Empl: Th vlocity of cclrtig cr is giv y v t ) 5 t ( m/s. 5 v(t) How fr os it trvl from t to t 6s? Sic istc is oti from v t 6 5
So 6 6 t v( t)t 5 t Not tht 5 t t 5 t t t Th, puttig ( t ) 5 t, 6 Distc ( 6) () 5 6 5( ) 499 m.499 km. This is th sis of Dfiit tgrls. F( ) Th rltioship tw f ( ) F( ) F( ) f ( t) t c grlis. 6
3. fiit tgrls W c fi itgrtio s th ivrs of iffrtitio: F( ) f f ( ), th th ifiit itgrl of f () with rspct to is fi y f ) F( ) C ( whr C is ipt of. prctic, th two forms of itgrtio giv th sm rsult, with f ) f ( t)t ( D, whr D is ipt of. (For fuctios of o vril, C D r costts, ut this coms ltr i th cours.) 3 Dfiit itgrls such s t t t giv vlu s rsult. fiit itgrls such s t t t t C giv fuctios 7
3.4 qulitis for tgrls () f ( ) f ( ) f() f() - () f ( ) g( ), if f ( ) g( ) for < <. g() f() 8
3.6 TECHNQUES FOR NTEGRATON Rcogitio f you kow iffrtitio, you kow itgrl / 3 / 3, 3 3 3.g. sic ( ) ( ) 3 3 / 3 3 Th ( ) ( ) / 3 C So r th tls of iffrtils i HLT p.8. ckwrs Sustitutio to Rcogisl Form.g. Put sihy, hc coshy y 9
sih y cosh y coshy y y coshy Th y c sih C T hlf gl formul - spcil sustitutio Tious ut usful: w fi t t θ th θ θ t sc θ t θ ( t )θ so θ t ( t ) Similrly θ t θ θ t siθ si cos or θ sc t t tθ t cosθ cos θ si θ ( t ( t ) )
.g. t t cos - t t t t t c t c Similrly, cot c cos Try t hlf gl formul wh ll ls fils
- SYMMETRY O Fuctios f ( ) f ( ) f Oviously ( ) yf() - Ev Fuctios f ( ) f ( ) yf() - So f ( ) f ( )
vrs Fuctios y y f () or f ( ) From th igrm So, f ( y)y y y f ( ) f ( y) y y c f ( ) ( crful with costts of itgrtio, i.. c ov) c y c c y f - (y) y f() f - (y) y f() Empl y ; ly Th l y y y c (l y) y y (l y ) c c 3
Prtil Frctios Sic l( ) C ( ) Prtil frctios c simplify itgrls..g. ( )( ) A B C D Put ( ) ( ) ( )( ) ( ) Multiplyig out th umrtor: A ( )( ) B( ) C( )( ) D( ) 4
5 Collctig powrs of : : 3 C B A A : D B A 4 B : D C B A 4 3 C : A D So ) ( ) 4( 3 ) 4( C ) ( ) l( 4 3 ) l( 4 l ( ) ( ) C ) ( l 4 3 / 4 /
6 tgrtio y Prts Th Prouct rul for iffrtitio, u v v u uv ) ( c rwritt s u v uv v u ) ( tgrtig u v uv v u ) ( i.. u v uv v u This c us to tur itgrl ito sir form.
Empl cos Put u, v cos, v si Th si si () Put v u, si si [ ( cos ) ()( cos ) ] si cos si C Tip: f t first you o t succ try it th othr wy rou 7
Rcursio Empl: cos (si) si si Try th othr wy rou?? [ ] cos cos si ( si ) cos Solutio : ( si cos ) This c us to grt Ructio Formul 8
9 Empl ( ) ( ) - ( ) ( ) Dos this hlp?
Sic C w c grt ll th othr :- C ) ( ( ) ( ) C ) ( ( ) C tc. Tip: Alwys chck y iffrtitig th swr
Furthr mpls: π π () si θ θ si π [ si θ cosθ ] ( ) π si θ ( ) ( ). θ (-cosθ )θ θ π ( si θ ) Hc. ( )si θ θ cos θ θ π m, m () (for you to try ) cos si. Show tht m, m,. m