Understanding and Evaluating the Integral of Euclidean Distance in a Unit Square

When evaluating integrals with a geometric interpretation, such as the Euclidean distance between two points, we can convert the integral into a more approachable form and leverage symmetry for simplification. In this article, we will systematically evaluate the integral within a unit square and explain the underlying mathematics and geometric principles.

Introduction

Consider the integral [ I int_0^1 int_0^1 int_0^1 int_0^1 sqrt{(x_2 - x_1^2)(y_2 - y_1^2)} , dx_1 , dx_2 , dy_1 , dy_2. ] This integral can be seen as the average Euclidean distance between two points in a unit square.

Step 1: Simplify the Integral

The expression ( sqrt{(x_2 - x_1^2)(y_2 - y_1^2)} ) can be interpreted as the distance between the points ( (x_1, y_1) ) and ( (x_2, y_2) ) in the plane. The integral is essentially an average of these distances over the unit square ( [0,1] times [0,1] ) for both points.

Step 2: Change of Variables

To simplify the evaluation, we consider the expected value of the distance between two random points in the unit square. This approach allows us to express the integral as: [ I int_0^1 int_0^1 int_0^1 int_0^1 dx_1 , dy_1 , x_2 , y_2 , dx_2 , dy_2, ] where ( dx_1 , dy_1 , x_2 , y_2 sqrt{(x_2 - x_1^2)(y_2 - y_1^2)} ).

Step 3: Consider Symmetry

Due to the symmetry of the problem, we can focus on the distances in one dimension and then extend it to the two dimensions. This symmetry simplifies the calculation and allows us to evaluate the integral in a systematic manner.

Step 4: Calculate the Integral

The Euclidean distance can be rewritten in terms of the joint distributions of ( x_1, x_2, y_1 ) and ( y_2 ).

Calculate the Inner Integral for ( x )

First, let's calculate the integral for ( x ): [ I_x int_0^1 int_0^1 sqrt{x_2 - x_1^2} , dx_1 , dx_2. ] We can split this integral based on the cases where ( x_2 geq x_1 ) and ( x_2 Similarly, for ( y ), the integral evaluates to the same value:

Calculate the Integral for ( y )

The calculation for ( y ) is identical to that for ( x ), thus:

[ I_y frac{1}{3}. ]

Step 5: Combine Results

Since the two dimensions are independent, we combine the results by multiplying them together:

[ I I_x cdot I_y frac{1}{3} cdot frac{1}{3} frac{1}{9}. ]

Final Result: Thus, the value of the integral is:

[ boxed{frac{1}{6}}. ]

This is the expected distance between two random points in the unit square, a well-known result in geometric probability.