1 Line Integrals of Scalar Functions

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1 MA Fll 2010 Worksheet VIII 13.2 nd Line Integrls of Sclr Functions There re (in some sense) four types of line integrls of sclr functions. The line integrls w.r.t. x, y nd z cn be plced under one umbrell, but they will be introduced seprtely t first nd I will record ll of them for sclr functions f of three vribles. Throughout, r(t) = x(t), y(t), z(t),, t b, is the vector eqution of smooth curve. This is equivlent to the curve given by prmetric equtions x = x(t), y = y(t),, z = z(t), t b. Remrk 1.1 (Friendly Reminder: The rc length element). One more friendly reminder bout the rc length element ds expressed in terms of the prmeteriztion: (dx ) 2 ( ) 2 ( ) 2 dy dz ds = r (t) dt = + + dt. dt dt dt Remrk 1.2. When the curve is plne curve, then there re no z terms in ny of the bove. The line integrls of sclr functions long curve re s follows. The line integrl of sclr function long w.r.t. rc length: b b f ds = f ( r(t)) r (t) dt = f (x(t), y(t), z(t)) r (t) dt. The line integrl of sclr function f long the curve w.r.t. x: f dx = b f ( r(t)) dx b dt dt = f (x(t), y(t), z(t)) dx dt dt. The line integrl of sclr function f long the curve w.r.t. y: f dy = b f ( r(t)) dy b dt dt = f (x(t), y(t), z(t)) dy dt dt. The line integrl of sclr function f long the curve w.r.t. z: f dz = b f ( r(t)) dy b dt dt = 1 f (x(t), y(t), z(t)) dz dt dt.

2 1.1 Problems 1. Let be the top hlf of the circle of rdius two in the xy-plne. A vector eqution for this curve is r(t) = 2 cos(t), 2 sin(t) with 0 t π. With this prmeteriztion, you trverse the curve counterclockwise from the point (2, 0) to the point ( 2, 0). () Evlute the line integrl with respect to rc length of the sclr function f(x, y) = xy 2 long the curve. Tht is, evlute xy 2 ds. (b) See the figure below nd interpret your nswer to 1.) geometriclly in terms of signed re. Explin why your nswer mkes sense. Figure 1: The plot of the curve (green) nd the grph of f(x, y) = xy 2 tht lies over the curve (blue) (i.e. the grph of the function restricted to the curve) in 2.) (c) Evlute the line integrl with respect to x of the sclr function f(x, y) = xy 2 long the curve. Tht is, evlute xy 2 dx. (d) Evlute the line integrl with respect to y of the sclr function f(x, y) = xy 2 long the curve. Tht is, evlute xy 2 dy. 2

3 Remrk 1.3. The integrtion in this problem ( 1d.) ) might be slightly tricky. 2. Let be the portion of the circle of rdius two tht lies in the first hlf of the xy-plne. A vector eqution for this curve is r(t) = 2 cos(t), 2 sin(t) with 0 t π. With this 2 prmeteriztion, you trverse the curve counterclockwise from the point (2, 0) to the point (0, 2). () Evlute the line integrl with respect to rc length of the sclr function f(x, y) = xy 2 long the curve. Tht is, evlute xy 2 ds. (b) See the figure below nd interpret your nswer to 2.) geometriclly in terms of signed re. Explin why your nswer mkes sense. Figure 2: The plot of the curve (green) nd the grph of f(x, y) = xy 2 tht lies over the curve (blue) (i.e. the grph of the function restricted to the curve) in 2.) 3. Let be the portion of the helix given by the vector eqution r(t) = 4t, 3 cos(t), 3 sin(t) with 0 t π. The initil point for this prmeteriztion is r(0) = 0, 3, 0. The terminl point for this prmeteriztion is r(π) = 4π, 3, 0. () Evlute the line integrl (w.r.t. rc length) ds. Interpret your nswer geometriclly. (Hint: Think of chpter 10 or think of wht the infinitesiml pieces being dded together re.) 3

4 (b) Evlute the line integrl with respect to rc length of the sclr function f(x, y, z) = yz cos(x) long the curve. Tht is, evlute yz cos(x) ds. 4. Evlute the line integrl xy z dx, where is the curve given by the vector eqution r(t) = t 3, t 2, t 4, with 0 t Evlute the line integrl xy dx + y 2 dy + yz dz, where is the line segment from (1, 0, -1) to (3, 4, 2). Note tht vector eqution for this curve is r(t) = (1 t) 1, 0, 1 + t 3, 4, 2 = 1 + 2t, 4t, 3t 1, with 0 t 1. 4

5 2 Line Integrls of Vector Fields Remrk 2.1. I m going to stte the generl formuls for the line integrl of vector field F(x, y, z) = P (x, y, z), Q(x, y, z), R(x, y, z) = P (x, y, z) i + Q(x, y, z) j + R(x, y, z) k on R 3 long curve. The curve is given the sme mnner s from the top of pge 1. The definitions on R 2 re nlogous (eliminte ll z components) The line integrl of vector field F(x, y, z) long curve cn be expressed in ny of the following forms (you should pick your fvorite, they re ll equl). If you try nd memorize ll of these, it will lmost certinly end in disster. F T ds = F d r = = b F ( r(t)) r (t) dt = = = b b b = F (x(t), y(t), z(t)) r (t) dt P (x(t), y(t), z(t)), Q(x(t), y(t), z(t)), R(x(t), y(t), z(t)) dx dt, dy dt, dz dt dt P (x(t), y(t), z(t)) dx dy dz dt + Q (x(t), y(t), z(t)) dt + R (x(t), y(t), z(t)) dt dt dt dt P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz 5

6 1. Let F(x, y) = x 2 y 3, y x nd let be the curve given by the prmetric equtions x = t 2, y = t 3, 0 t 1. () Evlute the line integrl of the vector field F long the curve. (b) See the figure below nd explin why the sign of your nswer mkes sense (either geometric interprettion or physicl interprettion). Figure 3: The plot of the vector field F (blue) nd the curve (red) in problem 1. Be creful, direction of trvel long the curve mtters. 2. Let F be vector field on R 2 defined by F(x, y) = ln(1 + y 2 ), ln(1 + x 2 ). Let be the portion of the circle of rdius 4 in the second qudrnt given by the prmetric equtions 3π x = 4 sin(t), y = 4 cos(t), 2 t 2π. Note tht with this prmetiztion you re trversing the circle in the clockwise direction from the initil point (-4, 0) to the terminl point (0, 4). () Evlute the line integrl of F long the curve (Hint: You should use Mple or your grphing clcultor to evlute the definite integrl once everything is in terms of t.) (b) See the figure below nd explin why the sign of your nswer mkes sense (either geometric interprettion or physicl interprettion). 6

7 Figure 4: The plot of the vector field F (blue) nd the curve (red) in problem 2. Be creful, direction of trvel long the curve mtters. (c) Wht is the vlue of the line integrl of F(x, y) long the sme curve if the curve is trversed in the opposite direction? (Tht is, if insted one trvels from (0,4) to (-4, 0)). Explin this geometriclly (or physiclly). You should not need to integrte. 3. Let F(x, y, z) = yx, zy, xz nd let be the curve given by the vector eqution r(t) = t 3, t 2, t, with 0 t 1. Evlute the line integrl of the vector field F(x, y, z) long the curve. 4. Find the work done by the force field F(x, y, z) = y + z, x + z, x + y on prticle tht moves long the line segment from (1, 0, 0) to (3, 4, 2). Note tht the prmetric equtions of this curve re x = 1 + 2t, y = 4t, z = 2t, 0 t 1. Hint: Think of our physicl motivtion for the definition of the line integrl of vector field (Or just think bout the line integrl of vector field!). 3 Fundmentl Theorem for Line Integrls Remrk 3.1. All problems from this section depend on the mteril from 13.3 in your text. You need to know the fundmentl theorem for line integrls, the definition of conservtive vector field, the definition of potentil function, nd the reltionships between the three. The mjority of the problems in this section will tke little work if done efficiently. 7

8 1. Let f(x, y) = y 3 x 2 + x 2 y + 2. omplete ech of the following. () Wht is the grdient of f(x, y)? (b) Is the vector field in the previous prt conservtive? If yes, wht is the potentil function? (c) Evlute the line integrl of the vector field f(x, y) long the curve given by the vector eqution r(t) = t, t 2 + 2t 3, 0 t 2. (d) How much did the given curve in the preceding prt mtter in obtining your finl nswer? (e) Evlute the line integrl of the vector field f(x, y) long the unit circle, where the unit circle is given the prmeteriztion x = cos(t), y = sin(t), 0 t 2π. Note tht this curve is one trip round the unit circle in counter-clockwise direction. (f) Evlute the line integrl of the vector field f(x, y) long curve from the point (1,1) to (2, 2). 2. (To hmmer home the point...) Let f(x, y) = cos(y)e sin(x) + sin(y) cos(x). omplete ech of the following. () Wht is the grdient of f(x, y)? (b) Is the vector field in the previous prt conservtive? If yes, wht is the potentil function? (c) Evlute the line integrl of the vector field f(x, y) long the curve given by the vector eqution r(t) = t, t 2 + 2t 3, 0 t 2. (d) How much did the given curve in the preceding prt mtter in obtining your finl nswer? (e) Evlute the line integrl of the vector field f(x, y) long the unit circle, where the unit circle is given the prmeteriztion x = cos(t), y = sin(t), 0 t 2π. Note tht this curve is one trip round the unit circle in counterclockwise direction. (f) Evlute the line integrl of the vector field f(x, y) long curve from the point (0, 0) to ( π 2, 2π). 8

9 3. Let F(x, y) = y, x. () Evlute the line integrl of the vector field F(x, y) long the unit circle, where the unit circle is given prmeteriztion x = cos(t), y = sin(t), 0 t 2π. Note tht this curve is one trip round the unit circle in counterclockwise direction. Explin the geometry (or physics) of your nswer in 3.) (b) Explin wht your nswer from the previous prt tells you bout the vector field F(x, y). Figure 5: The plot of the vector field F blue nd the curve in problem 3. (c) If you hve concluded tht the vector field F is not conservtive, then given second justifiction for this. If you hve concluded tht the vector field F is conservtive, then find sclr potentil for F. 4. The vector field F(x, y) = 2xy, x 2 is conservtive. () Find sclr potentil for F(x, y). (b) Use the sclr potentil from the previous prt to evlute the line integrl of F(x, y) long the curve given by the prmetric equtions x = e t cos(t), y = sin(t), 0 t π The vector field F(x, y) = (1 + xy)e xy, e y + x 2 e xy is conservtive. 9

10 () Find sclr potentil for F(x, y). (b) Use the sclr potentil from the previous prt to evlute the line integrl of F(x, y) long the curve given by the prmetric equtions x = t 2 + 1, y = t 3, 0 t The vector field F(x, y, z) = y + z, x + z, x + y is conservtive. () Find sclr potentil for F(x, y, z). (b) Use the sclr potentil from the previous prt to evlute the line integrl of F(x, y, z) long the curve given by the prmetric equtions x = t 2 + 1, y = t 3, z = e t, 0 t 1. 10