Use Green's Theorems to evaluate oint_C xydx + x^2y^3dy, where C is the triangle with vertices(0,0),(1,0)and (1,2).

Falak Kinney 2020-10-18 Answered
Use Green's Theorems to evaluate Cxydx+x2y3dy, where C is the triangle with vertices(0,0),(1,0)and (1,2).
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Expert Answer

diskusje5
Answered 2020-10-19 Author has 82 answers

Step 1
Consider the provided question,
Cxydx+x2y63dy, where c is the triangle with vertices (0,0),(1,0) and (1,2)
We know that, Green's theorem,
CP(x,y)dx+Q(x,y)dy=D(QxPy)dxdy
here, P(x,y)=xy,Q(x,y)=x2y3
and D is the region inside the triangle,
Now,
Py=x,Qx=2xy3
(QxPy)=2xy3x=x(2y31)
Step 2
We can describe the triangle D by, 0x1,0y<+2x
Cxydx+x2y3dy=x=01y=0xx(2y31)dxdy
=x=01xdxy=0x(2y31)dy
=x=01xdx[2y44y]0x
=x=01[x55x2]dx
=[x612x33]01
=11213
=14

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