14.2 Line Integrals. determines a partition P of the curve by points Pi ( xi, y

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1 4. Le Itegrls I ths secto we defe tegrl tht s smlr to sgle tegrl except tht sted of tegrtg over tervl [ ] we tegrte over curve. Such tegrls re clled le tegrls lthough curve tegrls would e etter termology. Le tegrls ple We strt wth ple curve gve y the prmetrc equtos () x x( t) y y( t) t or equvletly y the vector equto r( t) x( t) y( t) j t. A prtto of the prmeter tervl [ ] y pots t wth t t t determes prtto P of the curve y pots P ( x y ) where x x( t) d y y( t). These pots P dvde to surcs wth legths s s s. The orm P of the prtto s the logest of these legths. We choose y pot * * P ( x y ) the th surc. (Ths correspods to pot t [ t t].) Now f f s y fucto of two vrles whose dom cludes the curve we evlute f t ( x y ) multply y the legth f x y s ( ) s of the surc d form the sum whch s smlr to Rem sum. The we tke the lmt of these sums d mke the followg defto y logy wth sgle tegrl. If f s defed o curve gve y () the the le tegrl of f log s ( ) lm ( ) P f x y ds f x y s f the lmt exsts. Smlrly we c defe le tegrls of f wth respect to x d y : ( ) lm ( ) P f x y dx f x y x ( ) lm ( ) P f x y dy f x y y

2 We hve the followg formuls: Gve : x x( t) y y( t) t. dx dy f ( x y) ds f ( x( t) y( t)) f ( x y) dx f ( x( t) y( t)) x '( t) f ( x y) dy f ( x( t) y( t)) y'( t) Remrk : legth of curve Remrk : It frequetly hppes tht le tegrls wth respect to x d y occur together. Whe ths hppes t s customry to revte y wrtg P( x y) dx Q( x y) dy P( x y) dx Q( x y) dy Exmple : Evlute x y ds where s the upper hlf of the ut crcle ( ) x y. Soluto: The upper hlf ut crcle c e prmetrzed y mes of the equto: x cos t y s t t. ( ) ( cos s ) dx dy x y ds t t = ( cos ts t) s t cos t = ( cos ts t) t cos t Suppose ow tht s pecewse-smooth curve: tht s s uo of fte umer of smooth curves where s llustrted the followg Fgure. The we defe the tegrl of f log s the sum of the tegrls of f log ech of the smooth peces of : f ( x y) ds f ( x y) ds f ( x y) ds f ( x y) ds

3 Exmple : Evlute xds where cossts of the rc of the prol () followed y the vertcl le segmet from () to (). y x from () to Soluto: The prmetrc equto for s dx dy xds x dx dx dx = x 4 x dx 5 5 4x 4 x x y x x. O we c choose y s the prmeter the prmetrc equto s x y y y. dx dy xds () dy= dy dy dy 5 5 Thus xds xds xds Whe we re settg up le tegrl sometmes the most dffcult thg s to thk of prmetrc represetto for curve whose geometrc descrpto s gve. It s useful to rememer tht prmetrc represetto of le segmet ple strtg t ( x y ) d edg t ( x y ) s x ( t) x tx y ( t) y ty t [ le segmet spce strtg t ( x y z ) d edg t ( x y z ) s x ( t) x tx y ( t) y ty z ( t) z tz t Exmple : Evlute y dx xdy where () s the le segmet from (-5-) to () d () s the rc of the prol x 4 y from (-5-) to (). () A prmetrc represetto for the le segmet s x ( t)( 5) t() 5t 5 y ( t)( ) t() 5t t y dx xdy (5t ) (5 ) (5t 5)(5 ) 5 t t =5 (5 5 4)

4 () Sce the prol s gve s fucto of y let s tke y s the prmeter d wrte s Le Itegrls Spce x y y y y 4 - y dx xdy y ( y) dy (4 y ) dy = ( y y 4) 4 We hve the followg formuls: Gve : x x( t) y y( t) z z( t) t. 5 dx dy dz f ( x y z) ds f ( x( t) y( t) z( t)) f ( x y z) dx f ( x( t) y( t) z( t)) x '( t) f ( x y z) dy f ( x( t) y( t) z( t)) y '( t) f ( x y z) dz f ( x( t) y( t) z( t)) z '( t) Exmple 4: Evlute ydx zdy xdz where cossts of the le segmet from () to (45) followed y the vertcl le segmet from (45) to (4). Soluto: The prmetrc equto for s x ( t)() t() t y ( t)() t(4) 4 t z ( t)() 5( t) 5 t t ydx zdy xdz (4 t) (5 t)4 ( t)5 ( 9 t) 4.5 The prmetrc equto for s x ( t)() t() y ( t)(4) t(4) 4 z ( t)(5) ( t) 55 t t The dx dy sce x y 4 ydx zdy xdz ( 5) 5 ydx zdy xdz ydx zdy xdz ydx zdy xdz

5 Le Itegrls of Vector Felds Defto: Let F e cotuous vector feld defed o smooth curve gve y vector fucto rt ( ) t. The the le tegrl of F log curve s F dr F( r( t)) r '( t) Remrk: The work doe y the force feld F movg prtcle log the curve : r( t) t s Work = F dr Exmple 5: Fd the work doe y the force feld curve : r( t) cos t s t j t. F( x y) x xyj movg prtcle log the Soluto: Sce : r( t) cos t s t j t x cos t y s t thus F( r( t)) cot t cost s t j r '( t) s t cos t j Therefore the work doe s ( ( )) '( ) ( cos s ) cos t F dr F r t r t t t Remrk: If F P Qj Rk P Q R the we hve F dr Pdx Qdy Rdz