Method -I:-

Calculation of the Integral:

The given integral is:

1/(2π) ∫∫D (x, y, 10) dx dy

where D denotes the disc: x^2 + y^2 ≤ 10.

Change of Variables:
To simplify the integral, we can introduce a change of variables.

Let's consider the transformation:

u = x
v = y

The Jacobian determinant of this transformation is equal to 1, as there is no scaling or rotation involved.

Transformation of the Integral:
Using the change of variables, we can express the given integral in terms of u and v:

1/(2π) ∫∫D (x, y, 10) dx dy = 1/(2π) ∫∫D (u, v, 10) |J| du dv

where |J| is the determinant of the Jacobian matrix, which is equal to 1 in this case.

Integration Limits:
The region of integration D can be described in terms of the transformed variables u and v:

u^2 + v^2 ≤ 10

This represents a disc centered at the origin with radius √10.

Double Integral:
The integral can now be written as:

1/(2π) ∫∫D (u, v, 10) |J| du dv

1/(2π) ∫∫D (u, v, 10) du dv

The integral is separable, so we can evaluate it in two parts:

1/(2π) ∫∫D u du dv + 1/(2π) ∫∫D v du dv

Integration of u:
The integral 1/(2π) ∫∫D u du dv represents the average value of u over the region D, which is 0 since the region is symmetric about the origin.

Integration of v:
The integral 1/(2π) ∫∫D v du dv represents the average value of v over the region D.

Since the region D is symmetric about the x-axis, the average value of v over D is 0.

Final Result:
Therefore, the integral simplifies to:

1/(2π) ∫∫D u du dv + 1/(2π) ∫∫D v du dv = 0 + 0 = 0

Hence, the value of the integral is 0.