Jim Lmbers MAT 28 Spring Semester 29- Lecture 7 Notes These notes correspond to Section 3.2 in Stewrt nd Section 7.2 in Mrsden nd Tromb. Line Integrls Recll from single-vrible clclus tht if constnt force F is pplied to n object to move it long stright line from x to x b, then the mount of work done is the force times the distnce, W F (b ). More generlly, if the force is not constnt, but is insted dependent on x so tht the mount of force pplied when the object is t the point x is given by F (x), then the work done is given by the integrl W b F (x) dx. This result is obtined by pplying the bsic formul for work long ech of n subintervls of width Δx (b )/n, nd tking the limit s Δx. Now, suppose tht force is pplied to n object to move it long pth trced by curve, insted of moving it long stright line. If the mount of force tht is being pplied to the object t ny point p on the curve is given by the vlue of function F (p), then the work cn be pproximted by, s before, pplying the bsic formul for work to ech of n line segments tht pproximte the curve nd hve lengths Δs, Δs 2,..., Δs n. The work done on the ith segment is pproximtely F (p i )Δs i, where p i is ny point on the segment. By tking the limit s mx Δs i, we obtin the line integrl n W F (p) ds lim F (p i ) Δs i, mx Δs i provided tht this limit exists. In order to ctully evlute line integrl, it is necessry to express the curve in terms of prmetric equtions. For concreteness, we ssume tht is plne curve defined by the prmetric equtions x x(t), y y(t), t b. Then, if we divide [, b] into subintervls of width Δt (b )/n, with endpoints [t i, t i ] where t i + iδt, we cn pproximte by n line segments with endpoints (x(t i ), y(t i )) nd (x(t i ), y(t i )), for i, 2,..., n. From the Pythgoren Theorem, it follows tht the ith segment hs length (Δxi ) 2 ( ) 2 Δs i Δx 2 i + Δy2 i Δyi + Δt, Δt Δt i
where Δx i x(t i ) x(t i ) nd Δy i y(t i ) y(t i ). Letting Δt, we obtin b (dx ) 2 ( ) dy 2 F (p) ds F (x(t), y(t)) + dt. dt dt We recll tht if F (x, y), then this integrl yields the rc length of the curve. Exmple (Stewrt, Section 3.2, Exercise 8) To evlute the line integrl x 2 z ds where is the line segment from (, 6, ) to (4,, 5), we first need prmetric equtions for the line segment. Using the vector between the endpoints, we obtin the prmetric equtions It follows tht v 4, 6, 5 ( ) 4, 5, 6, x 4t, y 6 5t, z + 6t, t. x 2 z ds 6 77 (x(t)) 2 z(t) [x (t)] 2 + [y (t)] 2 + [z (t)] 2 dt (4t) 2 (6t ) 4 2 + ( 5) 2 + 6 2 dt 6t 2 (6t ) 77 dt 6t 3 t 2 dt 6 ( ) 77 6 t4 4 t3 3 6 ( 3 77 2 ) 3 56 77. 3 Exmple (Stewrt, Section 3.2, Exercise ) We evlute the line integrl (2x + 9z) ds 2
where is defined by the prmetric equtions x t, y t 2, z t 3, t. We hve (2x + 9z) ds (2x(t) + 9z(t)) [x (t)] 2 + [y (t)] 2 + [z (t)] 2 dt (2t + 9t 3 ) 2 + (2t) 2 + (3t 2 ) 2 dt (2t + 9t 3 ) + 4t 2 + 9t 4 dt 4 4 2 4 3 u3/2 u /2 du, u + 4t 2 + 9t 4 4 6 (43/2 ). Although we hve introduced line integrls in the context of computing work, this pproch cn be used to integrte ny function long curve. For exmple, to compute the mss of wire tht is shped like plne curve, where the density of the wire is given by function ρ(x, y) defined t ech point (x, y) on, we cn evlute the line integrl m ρ(x, y) ds. It follows tht the center of mss of the wire is the point ( x, y) where x xρ(x, y) ds, y yρ(x, y) ds. m m Now, suppose tht vector-vlued force F is pplied to n object to move it long the pth trced by plne curve. If we pproximte the curve by line segments, s before, the work done long the ith segment is pproximtely given by W i F(p i ) [T(p i )Δs i ] where p i is point on the segment, nd T(p i ) is the unit tngent vector to the curve t this point. Tht is, F T F cos θ is the mount of force tht is pplied to the object t ech point on the 3
curve, where θ is the ngle between F nd the direction of the curve, which is indicted by T. In the limit s mx Δs i, we obtin the line integrl of F long, F T ds. If the curve is prmetrized by the the vector eqution r(t) x(t), y(t), where t b, then the tngent vector is prmetrized by T(t) r (t)/ r (t), nd, s before, ds [x (t)] 2 + [y (t)] 2 dt r (t) dt. It follows tht F T ds b F(r(t)) r (t) r (t) r (t) dt b F(r(t)) r (t) dt F dr. The lst form of the line integrl is merely n bbrevition tht is used for convenience. As with line integrls of sclr-vlued functions, the prmetric representtion of the curve is necessry for ctul evlution of line integrl. Exmple (Stewrt, Section 3.2, Exercise 2) We evlute the line integrl F dr where F(x, y, z) z, y, x nd is the curve defined by the prmetric vector eqution r(t) x(t), y(t), z(t) t, sin t, cos t, t π. We hve π F dr F(r(t)) r (t) dt π z(t), y(t), x(t) x (t), y (t), z (t) dt π cos t, sin t, t, cos t, sin t dt π [cos t + sin t cos t + t sin t] dt π π π cos t dt + sin t cos t dt + t sin t dt sin t π + π π 2 sin2 t t cos t π + cos t dt π. 4
If we write F(x, y) P (x, y), Q(x, y), where P nd Q re the component functions of F, then we hve b F dr F(r(t)) r (t) dt b b P (x(t), y(t)), Q(x(t), y(t)) x (t), y (t) dt P (x(t), y(t))x (t) dt + b Q(x(t), y(t))y (t) dt. When the curve is pproximted by n line segments, s before, the difference in the x-coordintes of ech segment is, by the Men Vlue Theorem, where t i t i t i. For this reson, we write nd conclude Δx i x(t i ) x(t i ) x (t i ) Δt, b b P (x(t), y(t))x (t) dt Q(x(t), y(t))y (t) dt F dr P dx + Q dy. P dx, Q dy, These line integrls of sclr-vlued functions cn be evluted individully to obtin the line integrl of the vector field F over. However, it is importnt to note tht unlike line integrls with respect to the rc length s, the vlue of line integrls with respect to x or y (or z, in 3-D) depends on the orienttion of. If the curve is trced in reverse (tht is, from the terminl point to the initil point), then the sign of the line integrl is reversed s well. We denote by the curve with its orienttion reversed. We then hve F dr F dr, nd P dx P dx, Q dy Q dy. All of this discussion generlizes to spce curves (tht is, curves in 3-D) in strightforwrd mnner, s illustrted in the exmples. 5
Exmple (Stewrt, Section 3.2, Exercise 6) Let F(x, y) sin x, cos y nd let be the curve tht is the top hlf of the circle x 2 + y 2, trversed counterclockwise from (, ) to (, ), nd the line segment from (, ) to ( 2, 3). To evlute the line integrl F T ds sin x dx + cos y dy, we consider the integrls over the semicircle, denoted by, nd the line segment, denoted by 2, seprtely. We then hve sin x dx + cos y dy sin x dx + cos y dy + sin x dx + cos y dy. 2 For the semicircle, we use the prmetric equtions x cos t, y sin t, t pi. This yields sin x dx + cos y dy π sin(cos t)( sin t) dt + cos(sin t) cos t dt cos(cos t) π + sin(sin t) π cos( ) + cos(). For the line segment, we use the prmetric equtions x t, y 3t, t. This yields 2 sin x dx + cos y dy sin( t)( ) dt + cos(3t)(3) dt We conclude cos( t) + sin(3t) cos( 2) + cos( ) + sin(3) sin() cos(2) + cos() + sin(3). sin x dx + cos y dy cos() cos(2) + sin(3). In evluting these integrls, we hve tken dvntge of the rule b f (g(t))g (t) dt f(g(b)) f(g()), 6
from the Fundmentl Theorem of lculus nd the hin Rule. However, this shortcut cn only be pplied when n integrl involves only one of the independent vribles. Exmple (Stewrt, Section 3.2, Exercise 2) We evlute the line integrl F dr where F(x, y, z) P (x, y, z), Q(x, y, z), R(x, y, z) z, x, y, nd is defined by the prmetric equtions x t 2, y t 3, z t 2, t. We hve F dr 3 2. P dx + Q dy + R dz z(t)x (t) dt + x(t)y (t) dt + y(t)z (t) dt t 2 (2t) dt + t 2 (3t 2 ) dt + t 3 (2t) dt 2t 3 dt + 3t 4 dt + 2t 4 dt (5t 4 + 2t 3 ) dt ( 5 t5 5 + 2t4 4 ) 7
Prctice Problems Prctice problems from the recommended textbooks re: Stewrt: Section 3.2, Exercises -3 odd, 7, 9, 27, 29, 33, 35 Mrsden/Tromb: Section 7.2, Exercises, 3, 7, 9 8