# Line integral of a vector field along a curve C with two segments

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!

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

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 $$I=I_1+I_2$$