Use Green's Theorem to evaluate int_C vec(F)*d vec(r) where vec(F)(x,y)=xy^2 i + (1-xy^3)j and C is the parallelogram with vertices (-1,2), (-1,-1),(1,1)and(1,4). The orientation of C is counterclockwise.

fortdefruitI

fortdefruitI

Answered question

2021-03-01

Use Green's Theorem to evaluate CFdr where F(x,y)=xy2i+(1xy3)j and C is the parallelogram with vertices (-1,2), (-1,-1),(1,1)and(1,4).
The orientation of C is counterclockwise.

Answer & Explanation

bahaistag

bahaistag

Skilled2021-03-02Added 100 answers

Step 1
Using Green theorem the integral becomes:
Cxy2dx+(1xy3)dy=S(x(1xy3)yxy2)dxdy
where S is the surface enclosed by C.
Step 2
Thus, solve the integral noting that xyx+3and1x1:
Cxy2dx+(1xy3)dy=S(x(1xy3)yxy2)dxdy
=S(y32xy)dxdy
=x=11y=xx+3(y32xy)dxdy
=x=11((x+3)4x44x((x+3)2x2))dx
=x=1112x378x2144x814dx
Step 3
Neglecting all odd powers of x:
Cxy2dx+(1xy3)dy=x=1178x2814dx
=14x=1178x281dx
=12(78381)
=1072
Step 4
Thus, the integral is 1072=53.5.

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