LINE INTEGRALS AND SURFACE INTEGRALS TEST 5 HOMEWORK PACKET A. Vector Fields and Line Integrals 1. Match the vector field with its plot.

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1 LINE INTEGRALS AND SURFAE INTEGRALS TEST 5 HOMEWORK PAKET A. Vector Field and Line Integral 1. Match the vector field with it plot. F( y, ) = i+ ( y) F(, y) = yi + y F( y, ) = i+ d) F(, y) = co( + y) i + y + y + y. Determine the gradient vector field for the function and ketch a few vector from it. fy (, ) = y f ( y, ) = + y

2 Line Integral of Scalar Function 3. Evaluate the following line integral (d type) where i the given curve. yd, : = t, y= t, 0 t 1 4 y d, i the right half of the circle + y = 16 yz d, i the line egment from ( 1,5,0) to (1,6,4) d) ( + y + z ) d, : t =, y= cot, z= int, 0 t π 4. Evaluate the following line integral ( d, dy, dz type) 3 ( y ) dy, i the arc of the curve y= from (1,1) to (4,) z ye d, : = t, y t =, z 3 = t, 0 t 1 yd + zdy + dz, : = t, y = t, z = t, 1 t 4 d) z d + dy + y dz, i the line egment from (1,0,0) to (4,1,)

3 Line Integral through Vector Field 5. Evaluate the line integral F i d r (calculating work) F(, y) = ( y) i + ( ) : ( ) r t = ti + t, 0 t 1 1 F(, y) = e i + y : 3 r() t = t i + t, 0 t 1 F(, yz, ) = ini + coy + zk : 3 r() t = t i + t + tk, 0 t 1 6. The figure below how three curve, 1, and 3, lying in a vector field F. Are the line integral for 1, and 3 poitive, negative or zero? Eplain.

4 B. Fundamental Theorem for Line Integral (a.k.a. Line Integral in onervative Vector Field) and Green Theorem 1. Determine whether or not the vector field i conervative. If it i, find the function f uch that f =F (the calar function f i called the potential function. ) F(, y) = e inyi + e coy F(, y) = e coyi + e iny 3 F(, y) = (lny+ y ) i + (3 y + / y). The figure below how the vector field connecting the point (1,) and (4,3). F(, y) = y i+ y along with three curve Eplain why F i d r ha the ame value for all three curve. What i the integral value? 3. Find the potential function for the vector field F and ue it to evaluate 3 F(, y) = (lny+ y ) i + (3 y + / y) i the line egment from (,5) to (5,) Fi d r F(, y) = (3+ e in y) i + e coy : r( t) = coti+ int, 0 t π / F(, y, z) = ( y z + z ) i+ yz + ( y + z) k : r() t = ti+ ( t+ 1) + t k, 0 t 1 d) F(, yz, ) = in yi+ ( co y+ co z) yinzk : r( t) = inti+ t+ tk, 0 t π /

5 Green Theorem 4. Evaluate thee line integral 1 t : The direct way you already know nd : Uing Green Theorem Pd+ Qdy in two way: ( yd ) + ( + ydy ) i the circle of radiu 3 centered at the origin. ounterclockwie orientation. y d + ydy = + + i graphed at the right Ue Green Theorem to evaluate the following line integral along the given curve. 1 4 tan (co + ) + ( in ln(ec )) y e d y y dy i the rectangle with vertice at (0,0), (,0), (, π /), (0, π /) lockwie orientation. y 4 d+ 3 y dy i the ellipe + y = ounterclockwie orientation. y d y dy (1 ) + ( + co( )) = i graphed at the right The arc are circular.

6 . Parametric Surface / Surface Area 1. Match the parametric urface with their graph. r( uv, ) = ui+ v+ ( + u+ v) k r( uv, ) = in( u) i+ 3co( u) + vk; 0 u π, 0 v I. r( uv, ) = vi+ u+ ( u v) k; 0 uv, 1 d) r( uv, ) = uinvi+ u + ucovk e) r( uv, ) = uinvi+ u+ ucovk II. III. V. IV.. Find a parametric repreentation for the urface. The portion of the plane z= 5 y lying inide the cylinder + y = 9 The portion of the phere of the z plane. + y + z = 16 lying on the poitive y ai ide The part of the cylinder y + z = 4 between the plane = 1 and = 5.

7 3. Determine the urface area for the following parametric urface. The portion of the plane with the parametric equation r( uv, ) = ( u+ v) i+ ( 3 u) + (1 + u v) k for 0 u and 1 v 1 The urface with the parametric equation 1 r( uv, ) = u i+ uv+ v k for 0 u 1 and 0 v The helicoid (cork crew) r( uv, ) = ucovi+ uinv+ vk for 0 u 1 and 0 v π (HINT: You ll need to refer to your integral table to integrate thi one.) 4. Determine the urface area for the following function urface. (POLAR converion may be needed ometime.) The firt octant portion of the plane + 3y+ z= 6. The portion of the urface z= y lying inide the cylinder + y = 9 The portion of the urface z= y lying above the triangle formed by the point (0,0), (0,1) and (,1). (HINT: ue d dy integration order.) d) The part of the phere cylinder + y + z = b lying above the y plane and inide the + y = a, where a< b.

8 D. Surface Integral and Flu r r r 1. alculate f ( yzds,, ) = f( ( uv, )) u v da S D yz ds, S i the cone with vector equation r( uv, ) = ucovi+ uinv+ uk for 0 u 1 and 0 v π / yds, S i the helicoid with vector equation r( uv, ) = ucovi+ uinv+ vk for 0 u 1 and 0 v π + y ds, S i the urface with vector equation r( uv, ) = uvi+ ( u v) + ( u + v) k for u + v 1. alculate S D f ( yzds,, ) = fygy (,, (, )) 1 + ( g) + ( g) ds y d) yz ds, S i the part of the plane 1 3 the rectangle [0,3] [0,] yds, S i the urface z ds z= + y for 0 1, S i the part of the cone plane z = 1 and z = 3. (HINT: co z= + + y lying above and 0 y z = + y lying between the θ = + coθ) 1 1 z+ y zds, S i the upper half of a phere of radiu centered at the origin.

9 3. alculate flu FindS = F(, y, g(, y)) i g, g,1 da S D F(, yz, ) = yi+ yz + zk, S i the part of the plane z= 6 lying above the quare [0,1] [0,1] with upward orientation. y F(, yz, ) = yi+ z+ k, S i the portion of the paraboloid above the y plane with upward orientation. z y = 1 lying F(, yz, ) = i z + yk, S i the part of the phere + y + z = 4 lying in the firt octant. Orientation i downward. E. Divergence Theorem 1. Find the divergence for the following vector field. div F = i F z 3 z F(, yz, ) = yei+ yz yek z F(, yz, ) = (co z+ y) i+ e + (in y+ z) k F(, yz, ) = inyi + co y zinyk 4 3 d) F(, yz, ) = i z + 4yzk. alculate the flu F S i d S acro the cloed urface uing the Divergence Theorem. Divergence Theorem: loed Surface Flu = S Fi ds= div FdV Where E i the olid region bounded by the cloed urface S. E z 3 z F(, yz, ) = yei+ yz yek S i the rectangular bo in the firt octant with one corner at (0,0,0) and the oppoite diagonal located at (3,,1) z F(, yz, ) = (co z+ y) i+ e + (in y+ z) k S i the urface of the olid bounded below by the paraboloid and bounded above by z = 4 F(, yz, ) = inyi + co y zinyk S i the oft cube + y + z = 56 z= + y d) 4 3 F(, yz, ) = i z + 4yzk S i the urface of the olid lying inide the cylinder above the y plane and below the plane z= + + y = 1,