We are tasked with evaluating the double integral:

$\int_1^2 \int_4^6 \frac{x}{y^2} \, dy \, dx$

Let's proceed step-by-step:

Step 1: Solve the inner integral with respect to $y$

The inner integral is:

$\int_4^6 \frac{x}{y^2} \, dy$

Since $x$ is treated as a constant with respect to $y$, we can factor it out:

$x \int_4^6 \frac{1}{y^2} \, dy$

Now, solve $\int \frac{1}{y^2} \, dy$. The antiderivative of $\frac{1}{y^2}$ is:

$-\frac{1}{y}$

So the inner integral becomes:

$x \left[ -\frac{1}{y} \right]_4^6 = x \left( -\frac{1}{6} + \frac{1}{4} \right)$

Simplifying the expression inside the parentheses:

$x \left( \frac{1}{4} - \frac{1}{6} \right) = x \left( \frac{3}{12} - \frac{2}{12} \right) = x \cdot \frac{1}{12}$

Thus, the result of the inner integral is:

$\frac{x}{12}$

Step 2: Solve the outer integral with respect to $x$

Now we compute the outer integral:

$\int_1^2 \frac{x}{12} \, dx$

We can factor out the constant $\frac{1}{12}$:

$\frac{1}{12} \int_1^2 x \, dx$

The antiderivative of $x$ is $\frac{x^2}{2}$, so:

$\frac{1}{12} \left[ \frac{x^2}{2} \right]_1^2 = \frac{1}{12} \left( \frac{2^2}{2} - \frac{1^2}{2} \right) = \frac{1}{12} \left( \frac{4}{2} - \frac{1}{2} \right)$

Simplifying the expression inside the parentheses:

$\frac{1}{12} \left( 2 - 0.5 \right) = \frac{1}{12} \times 1.5 = \frac{1.5}{12} = \frac{1}{8}$

Thus, the value of the double integral is:

$\boxed{0.125}$

The correct answer is option (c) $0.125$.

Would you like further details on any step?

Here are 5 related questions for practice:

1. How do you approach evaluating iterated integrals in general?
2. What are the steps to finding antiderivatives of rational functions?
3. How does changing the order of integration affect the evaluation of double integrals?
4. What is the physical interpretation of a double integral in terms of area or volume?
5. How can double integrals be solved using polar coordinates?

Tip: When evaluating double integrals, always check if it's possible to separate variables or simplify the expression by factoring out constants to reduce complexity.