Explanation:

To understand why the correct answer is option 'A', let's break down the problem step by step.

Step 1: Change of Variables

We are given a change of variables in the form:

x(u, v) = uv
y(u, v) = v/u

This means that for any point (u, v), we can determine the corresponding point (x, y) using the above equations.

Step 2: Changing the Integrand

We are asked to find the double integral of a function f(x, y). Using the change of variables, we need to express the integrand in terms of the new variables (u, v).

Let's denote the new function as g(u, v). We can express g(u, v) as follows:

g(u, v) = f(x(u, v), y(u, v))
= f(uv, v/u)

So, the integrand changes from f(x, y) to f(uv, v/u).

Step 3: Determining the Limits of Integration

To evaluate the double integral, we also need to determine the new limits of integration in terms of the new variables (u, v).

Let's say the original limits of integration for x are [a, b] and for y are [c, d]. Using the change of variables, we can determine the new limits of integration for u and v.

The new limits for u will depend on the values of x = uv. Since x = uv, we can solve for u as follows:

u = x/v

Since v is a constant in this case, u varies from a/v to b/v.

The new limits for v will depend on the values of y = v/u. Since y = v/u, we can solve for v as follows:

v = uy

Since y is a constant in this case, v varies from c/y to d/y.

Step 4: Evaluating the Integral

Now that we have determined the new limits of integration and the new integrand, we can evaluate the double integral.

∬f(x, y) dA = ∬g(u, v) |J| dA

Where |J| is the absolute value of the Jacobian determinant of the transformation.

In this case, the Jacobian determinant is given by:

|J| = |∂(x, y)/∂(u, v)| = |(∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u)|

Plugging in the given change of variables, we have:

|J| = |v - (u)(-v/u^2)| = |v + v/u| = |2v/u|

Now, we can evaluate the double integral using the new limits of integration and the new integrand.

∬f(x, y) dA = ∬g(u, v) |J| dA
= ∫[a/v, b/v] ∫[c/y, d/y] f(uv, v/u) |2v/u| dy du

Simplifying the integral, we get:

∬f(x, y) dA = 2 ∫[a/v, b/v]