Divergence, Curl, Line Integrals ((c) of M55 Upd)

...

8. Evaluate the following line integrals given the following vector fields along their corresponding

path C.

(a) F(x, y) = ; C is the quarter unit circle from (0,1) to (1,0). (b) F(x, y) = ; C is the first quadrant arc of the circle R(t) =

, (c) F(x, y) = ; C is the union of the line segment from (-1,0) to

(-1,1) and the parabola 9y = (x - 2)2 x ∈ [-1,5]. (d) F(x, y) = ; C is the union of C

1

, the segment from (0,0) to (6,4), C

2

, the segment from (6,4) to (4,0) and C

3

, the segment from (4,0) to (0,0).

(e) F(x, y) =

〈

xe2y x2 + 2

〉 ,e2y ln(x2 + 2)

; C is the circle R(t) = , t ∈

[π,2π].

V. Surface Integrals

1. Evaluate the surface integral

∫∫

S x2 that lies below the plane z = 2. /

1+4x + y2

, where S is the portion of the praboloid z = x2 +y2

2. Determine the flux of F(x,y,z) = across the portion of 2x + 2y + z = 2

in the first quadrant with an upward orientation.

3. Let Σ be the part of the parabolic cylinder z = 1 - x2 in the first octant bounded by the

coordinate planes and the plane y = 1. Evaluate the surface integral

∫∫

Σ

x dS

4. Let S be the portion of the paraboloid z = 4-x2-y2 which lies above the xy-plane. Evaluate

∫∫

S

√

1+4x2 4 - z

+ y2

dS

5. Using the positive orientation of the surface S : z = 4 - x2 - y2, find the flux of F(x,y,z) =

x 2

, y 2

,z > over the portion od S that lies above the xy-plane.

6. A circle surface u2 + S v2 has = 4. the Evaluate vector function the surface R(u, integral v) = u

ˆ i+v ∫∫

S

(x2 ˆ j +uv

+ y2 k ˆ

over + 1)1/2 the dS.

region D bounded by the

7. Let Σ be the triangular surface consisting of the part of the plane z + x = 1 in the first octant

cut by the plane x + y = 1. Evaluate the surface integral

∫∫

Σ

y dS.

Compiled from previous first long exams, Math 55 module and other reviewers Courtesy of kmboydon, mpona