I am trying to evaluate the line integral
∫
c
(2xy)dx+
2
x
+
2
y
dy
where C is the ellipse
2
x
25
+
2
y
16
=1
from (5,0) to (0,4).​
F
(x,y) = ∇f . [ (2xy)

i
+

2
x
+
2
y


j
] where ∇f =
∂f
∂x

i
+
∂f
∂y

j
​​
∂f
∂x
= 2xy,
∂f
∂y
=
2
x
+
2
y
Solving for the potential and then plugging in the end points gives
64
3
In[]:=
DSolve[{Grad[f[x,y],{x,y}]=={2xy,
2
x
+
2
y
}},f[x,y],{x,y}]
Out[]=
f[x,y]
2
x
y+
3
y
3
+

1

f[x_,y_]:=
2
x
y+
3
y
3
​​f[0,4]-f[5,0]
64
3
Function LineIntegrate[] using ImplicitRegion[] gives error
In[]:=
LineIntegrate{2xy,
2
x
+
2
y
},{x,y}∈ImplicitRegion
2
x
25
+
2
y
16
==1,{{5<=x<=0},{0<=y<=4}}
ImplicitRegion
:{5≤x≤0} is not a valid variable.
Out[]=
LineIntegrate{2xy,
2
x
+
2
y
},{x,y}∈ImplicitRegion
2
x
25
+
2
y
16
1,{{5≤x≤0},{0≤y≤4}}
Function LineIntegrate[] using ILine[] gives correct answer (Path Independent)
In[]:=
LineIntegrate[{2xy,
2
x
+
2
y
},{x,y}∈Line[{{5,0},{0,4}}]]
Out[]=
64
3