Evaluate int_C x^2y^2 dx + 4xy^3 dy where C is the triangle with vertices(0,0),(1,3), and (0,3). (a)Use the Green's Theorem. (b)Do not use the Green's Theorem.

Cabiolab 2021-02-25 Answered
Evaluate Cx2y2dx+4xy3dy where C is the triangle with vertices(0,0),(1,3), and (0,3).
(a)Use the Greens
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Expert Answer

Asma Vang
Answered 2021-02-26 Author has 93 answers

Step 1
Given:
cx2y2dx+4xy3dy
where c is the triangle with vertices (0,0),(1,3),(0,3)
Step 2
cx2y2dx+4xy3dy
co comparing with cPdx+Qdy we have
P=x2y2andQ=4xy3
By Green's Theorem
cx2y2dx+4xy3dy=D(QxPy)dA
=D(4y32x2y)dA
D=[(x,y):0x1,3xy3]
=013x3(4y32x2y)dydx
=01(y4x2y1)y=3xy=3dx
=01(819x272x4)dx
=(81x3x3725x5)01
=3185
Step 3
cx2y2dx+4xy3dy
c1:y=3x
dy=3dx
0x1
c2:y=3
dy=0
x from 1 to 0
c3:x=0
dx=0
y from 3 to 0
Hence
I=01x2(3x)2dx+4x

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