Josephood7

Use Green's Theorem to evaluate the line integral of F=⟨x3,7x⟩F=⟨x3,7x⟩

around the boundary of the parallelogram in the following figure (note the orientation).

 Let F⃗ =14xeyi⃗ +7x2eyj⃗ F→=14xeyi→+7x2eyj→ and G⃗ =14(x−y)i⃗ +7(x+y)j⃗ G→=14(x−y)i→+7(x+y)j→. Let CC be the path consisting of lines from (0,0)(0,0) to (3,0)(3,0) to (3,2)(3,2) to (0,0)(0,0). Find each of the following integrals exactly:

Suppose

and CC is the counter-clockwise oriented sector of a circle centered at the origin with radius 22 and central angle π/3π/3. Use Green's theorem to calculate the circulation of F⃗ F→ around CC.

Calculate ∫C((4x+2y)i⃗ +(3x+8y)j⃗ )⋅dr⃗ ∫C((4x+2y)i→+(3x+8y)j→)⋅dr→ where CC is the circular path with center (a,b)(a,b) and radius mm, oriented counterclockwise. Use Green's Theorem.