To evaluate the given double integral by changing the order of integration, let's first look at the expression you provided:

$I = \int_0^a \int_{y^2/a}^y \frac{y \, dx \, dy}{(a - x) \sqrt{a x - y^2}}$

Step 1: Understand the limits of integration

The outer integral has limits from $y = 0$ to $y = a$, while the inner integral has limits from $x = \frac{y^2}{a}$ to $x = y$. This describes a region in the $xy$-plane.

We need to find the new limits after changing the order of integration. First, we identify the boundaries of the region:

1. For $y$ values: The outer integral has $y$ running from 0 to $a$, so $0 \leq y \leq a$.
2. For $x$ values: The inner integral has $x$ running from $\frac{y^2}{a}$ to $y$, so for each fixed $y$, $x$ ranges from $\frac{y^2}{a}$ to $y$.

Step 2: Describe the region of integration

The region of integration can be understood as:

• $y$ ranges from $0$ to $a$,
• $x$ ranges from $\frac{y^2}{a}$ to $y$.

We now need to express this in terms of $x$ first, followed by $y$.

• From the equation $x = \frac{y^2}{a}$, we get $y = \sqrt{ax}$.
• From the equation $x = y$, we see that the upper boundary for $y$ is $x$, so $y = x$.

Thus, the region in terms of $x$ and $y$ can be rewritten as:

• $x$ ranges from 0 to $a$,
• For each $x$, $y$ ranges from $\sqrt{ax}$ to $x$.

Step 3: Set up the integral with changed order

The double integral with the order of integration changed becomes:

$I = \int_0^a \int_{\sqrt{ax}}^x \frac{y \, dy \, dx}{(a - x) \sqrt{a x - y^2}}$

Step 4: Evaluate the integral (outline)

Now that the order is changed, this integral is still quite complex due to the nature of the integrand. However, performing direct evaluation might involve advanced techniques such as substitution, simplification of the square root, and further analysis of the integral's form. If you'd like, I can guide you through an appropriate substitution or numerical evaluation for this integral.

Would you like to proceed with that, or would you like further clarification on any step of the process?

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Related Questions:

1. What is the process for determining the region of integration for a double integral?
2. How do you handle square roots in integrals, especially in the context of substitution?
3. What methods can be used for evaluating integrals with complicated denominators?
4. Can you explain how to change the order of integration in more detail?
5. What techniques help simplify integrals with rational expressions and square roots?

Tip: When changing the order of integration, sketching the region of integration helps clarify the boundaries and ensures you choose the correct limits for the new order.