Math Line Integrals I

Transcription

1 Mth Line Integrls I Peter A. Perry University of Kentucky November 16, 2018

2 Homework Re-Red Section 16.2 for Mondy Work on Stewrt problems for 16.2: 1-21 (odd), (odd), 49, 50 Begin Webwork D1

3 Unit IV: Vector lculus Lecture 35 Lecture 36 Lecture 37 Lecture 38 Lecture 39 Lecture 40 Vector Fields Line Integrls Line Integrls Fundmentl Theorem Green s Theorem url nd Divergence

4 Gols of the Dy Know how to compute line integrls of sclr function in the plne Know how to compute line integrls of sclr function in spce

5 Preview: Line Integrls Our next topic will be integrls of sclr functions nd vector functions over curves in the plne nd in spce. If is curve in the plne or in spce, we ll lern how to compute: f (x, y) ds, the integrl of sclr function over plne curve f (x, y, z) ds, the integrl of sclr function over spce curve F dr, the integl of vector function F(x, y) over plne curve F dr, the integrl of vector function F(x, y, z) over spce curve In ll cses, we ll reduce these to lculus I nd II type integrls by prmeterizing the curve. We ll lso lern how to compute integrls like f (x, y) dx f (x, y) dy

6 Prmeterizing Pths y y x x Prmeterize the following pths: 1. The first plnr pth shown on the left 2. The second plnr pth shown on the left 3. The pth connecting (0, 0, 0) to (1, 0, 1) 4. The pth connecting (1, 0, 1) to (1, 2, 0)

7 The Integrl of Sclr Function over Plne urve If is plne curve, the line integrl of f long is f (x, y) ds = lim n n f (xi, y i ) s i i=1 where we pproximte the curve by n line segments of length s i As prcticl mtter, if is prmeterized by (x(t), y(t)) for t b, so f (x, y) ds = ds = ( ) dx 2 ( ) dy 2 + dt dt dt b ( ) dx 2 ( ) dy 2 f (x(t), y(t)) + dt dt dt

8 The Integrl of Sclr Function over Plne urve if is prmeterized by (x(t), y(t)) for t b, then b ( ) dx 2 ( ) dy 2 f (x, y) ds = f (x(t), y(t)) + dt dt dt 1. Find (x/y) ds if is the curve x = t2, y = 2t for 0 t 3 2. Find xy 4 ds if is the right hlf of the circle x 2 + y 2 = 16

9 Line Integrls over Piecewise Smooth urves A curve is piecewise smooth if it is union of smooth curves 1,... n. Some exmples re shown t left. 1 1 If consists of sevel smooth components, then f (x, y) ds = n f (x, y) ds i=1 i 1 1 Notice tht ech of these curves hs n orienttion tht determines how the curve is prmeterized the prmeteriztion should follow the rrows. 1. Find xy ds if is the first curve shown t left.

10 Another Kind of Line Integrl For lter use, we ll lso need the line integrl of f with respect to x nd the line integrl of f with respect to y: b f (x, y) dx = f (x(t), y(t))x (t) dt b f (x, y) dy = f (x(t), y(t))y (t) dt 1. Find ex dx if is the rc of the curve x = y 3 from ( 1, 1) to (1, 1) 2. Find x2 dx + y 2 dy if is the rc of the circle x 2 + y 2 = 4 from (2, 0) to (0, 2) followed by the line segment from (0, 2) to (4, 3)

11 Summry of Line Integrls in the Plne If is prmeterized curve (x(t), y(t)) where t b: f (x, y) dx = f (x, y) dy = f (x, y) ds = b b b f (x(t), y(t))x (t) dt f (x(t), y(t))y (t) dt f (x(t), y(t)) (x (t) 2 + y (t) 2 dt

12 Applictions - enter of Mss A wire of mss m nd density ρ(x, y) long curve hs center of mss x = 1 xρ(x, y) ds m y = 1 yρ(x, y) ds m A thin wire hs the shpe of the first qudrnt prt of circle with center t the origin nd rdius. If the density of the wire is ρ(x, y) = kxy, find the mss nd center of mss of the wire.

13 Line Integrls in Spce If is spce curve (x(t), y(t), z(t)) where t b, then f (x, y, z) ds = b f (x(t), y(t), z(t)) (x (t)) 2 + (y (t)) 2 + (z (t)) 2 dt 1. Find (x2 + y 2 + z 2 ) ds if is the spce curve (x(t), y(t), z(t)) = (t, cos 2t, sin 2t) for 0 t 2π

14 More Line Integrls in Spce n you guess how to define f (x, y, z) dx, f (x, y, z) dy, nd f (x, y, z) dz? 1. Find (x + z) dx + (x + z) dy + (x + y) dz if consists of the line segments from (0, 0, 0) to (1, 0, 1) nd from (1, 0, 1) to (0, 1, 2) z (1, 2, 0) (1, 0, 1) (0, 0, 0) y x