Josephood7

Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine for the following vector fields F whether the line integrals ∫CF⋅dr are positive, negative, or zero and type P, N, or Z as appropriate.

Find ∫CF⃗ ⋅dr⃗ for F⃗ =8yi⃗ −(siny)j⃗ on the curve counterclockwise around the unit circle C starting at the point (1,0).

Evaluate the line integral ∫CF⋅d r∫CF⋅d r where F=⟨sinx,cosy,5xz⟩F=⟨sin⁡x,cos⁡y,5xz⟩ and CC is the path given by r(t)=(−3t3,−t2,−2t)r(t)=(−3t3,−t2,−2t) for 0≤t≤1

Evaluate the line integral ∫CF⋅d r∫CF⋅d r where F=⟨sinx,cosy,5xz⟩F=⟨sin⁡x,cos⁡y,5xz⟩ and CC is the path given by r(t)=(−3t3,−t2,−2t)r(t)=(−3t3,−t2,−2t) for 0≤t≤1