Question

F=(-x-y)i+(x+y)j, curve C is the counterclockwise path around the circle with radius 2 centered at (8,8)math

Answer 1

we can parameterize it as
x=8+2cos(t)
y=8+2sin(t)
ds=(dx,dy)=(-2sin(t),2cos(t))

F*ds=(8+2cos(t))2sin(t)+(8+2sin(t))2sin(t)+(8+2cos(t))2cos(t)
+(8+2sin(t))2cos(t)=
16sin(t)+4sin(t)cos(t)+16sin(t)+4sin^2(t)+
16cos(t)+4cos^2(t)+16cos(t)+4sin(t)cos(t)=
32sin(t)+32cos(t)+4+8sin(t)cos(t)

∫₀ ^2π32sin(t)+32cos(t)+4+8sin(t)cos(t) dt=8π

Answer 2

. (There are two parts; not all of the answers are zero.) Let F~
(x; y) =
cos(
p
x2+y2
)
p
x2+y2
(x~i + y~j). Evaluate
H
C
F~
d~r where (A) C : ~r(t) = (cos(2t);sin(4t)) with 0 t ; or (B) C : ~r(t) = (cos(2t);sin(
t
4
)) with
0 t

Answer 3

In two or three dimensions, a force can be described by a vector whose magnitude is the strength of the force and whose direction is the direction in whichthe force is pushing -- for example, the vector 2 i represents a force whose strength is two units and that is pushing from west to east. If an objectmoves under the influence of a force, F, and its motion is represented by the vector S as shown in the figure below

then the work done by the force is