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Vector field $ \vec F = (3x^2y^3+8x)\vec i + 3x^3y^2\vec j$, along a curve C consisting of two segments C$_1$ and C$_2$.

Line segment C$_1$ given by $y = 0$ and $0 ≤ x ≤ x_0$ and the line segment C$_2$ given by $x = x_0$ and $0 ≤ y ≤ y_0$.

I need help calculating the line integral of:

$V(x_0,y_0) = \int_0\vec F \cdot d\vec r = \int_C ((3x^2y^3+8x)dx + 3x^3y^2dy) $

The boundaries in the segments really throw me off, any help would be very much appreciated.

Thank you very much!

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  • $\begingroup$ Try visualizing this path: the first segment is horizontal, the second vertical. $\endgroup$ – amd Nov 20 '18 at 23:42
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hint

along $C_1 , dy=0$ gives

$$I_1=8\int_0^{x_0}xdx=4x_0^2$$

along $C_2, dx=0$ and

$$I_2=3x_0^3\int_0^{y_0}y^2dy=x_0^3y_0^3$$

the result is $$I=I_1+I_2$$

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