Potential Difference Over a Crazy Path

@Karan Chatrath has provided a nice summary of how to solve this problem. I will describe the curve generation. My approach is to just apply a random force to a simulated particle and plot the particle trajectory. This code changes the force every second, and runs the simulation for 100 seconds. It also normalizes the speed to unity on every processing interval to prevent the particle from speeding up. It is somewhat amusing to generate these curves and plot them. Here is another random curve, as an example (not the same as the one from the problem)

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import math
import random

##################################################

# Initialization

m = 1.0

x = 10.0
y = 10.0

theta = 2.0*math.pi*random.random()

xd = math.cos(theta)
yd = math.sin(theta)

Fx = -5.0 + 10.0*random.random()
Fy = -5.0 + 10.0*random.random()

xdd = Fx/m
ydd = Fy/m

dt = 10.0**(-4.0)

t = 0.0
count = 0

##################################################

print "x y"

while t <= 100.0:

    x = x + xd * dt
    y = y + yd * dt

    xd = xd + xdd * dt
    yd = yd + ydd * dt

    if count%(10**4) == 0:

        Fx = -5.0 + 10.0*random.random()
        Fy = -5.0 + 10.0*random.random()

        xdd = Fx/m
        ydd = Fy/m

    v = math.hypot(xd,yd)

    xd = xd/v
    yd = yd/v

    if count%(10**2) == 0:
        print x,y

    t = t + dt
    count = count + 1

Given:

$\vec{F} = \frac{x}{(x+y)\sqrt{x^2+y^2}} \hat{i} + \frac{y}{(x+y)\sqrt{x^2+y^2}} \hat{j}$ $d\vec{l} = dx \hat{i} + dy \hat{j}$

$d(\Delta U) = \vec{F} \cdot d\vec{l}$

Now, the numeric data is of length 10000. Let arbitrary index of the X-array and Y-array be denoted as $k$ where $k$ varies from 2 to 10000. $F_x(k) = \frac{x(k)}{(x(k)+y(k))\sqrt{x(k)^2+y(k)^2}}$ $F_y(k) = \frac{y(k)}{(x(k)+y(k))\sqrt{x(k)^2+y(k)^2}}$ $dx(k) = x(k) - x(k-1)$ $dy(k) = y(k) - y(k-1)$ $dI(k)=F_x(k)dx(k) + F_x(k)dy(k)$

Add up all elements of the column $dI$ to obtain the required answer of $\boxed{-0.27196}$ (work done using Excel). The path over which the integral is computed is as follows. To obtain a random path, I would just think of a random two-variable, time-varying (non-autonomous) and coupled non-linear system of differential equations (hopefully exhibiting chaotic behaviour) and solve them to obtain some arbitrary trajectory. But that thought process involves a bit of work. Is there a faster way?