Use Green's Theorem to evaluate int_C(e^x+y^2)dx+(e^y+x^2)dy where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by y = x^2 and y = 4.

CMIIh 2021-02-25 Answered
Use Green's Theorem to evaluate C(ex+y2)dx+(ey+x2)dy where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by y=x2andy=4.
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Expert Answer

Szeteib
Answered 2021-02-26 Author has 102 answers

Step 1
The give integral is C(ex+y2)dx+(ey+x2)dy ….(1)
where C is the boundary of the region in first quadrant bounded by y=x2andy=4.
From greens theorem we know that
CPdx+Qdy=D(QxPy)dxdy ....(2)
Comparing we get
P=ex+y2andQ=ey+x2
So,
Py=2yandQx=2x ...(3)
Step 2
From (1), (2) and (3) we get
C(ex+y2)dx+(ey+x2)dy=D2(xy)dxdy ...(4)
Now, we will find the region of integration.
y=x2andy=4
this gives
x2=4
x=+2
Since the region lies on first quadrant so we leave negative sign.
Therefore, x varies from 0 to 2 and y varies from 4 to x2
Now, from (4)
Step 4
C(ex+y2)dx+(ey+x2)dy=x=02y=4x22(xy)dxdy
=2x=02(xyy22)4x2dx
=2x=02(x×x2(x2)22)4x+8)dx
=2x=02(x3x424x+8)dx
=2(

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