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The image shows a double integral expression:

\[
\int_0^{\sqrt{2}} \int_0^x \frac{x}{\sqrt{x^2 + y^2}} \, dy \, dx
\]

It appears that the goal is to change the order of integration.

### To change the order of integration, we first need to examine the integration region:
- The limits for \( x \) go from \( 0 \) to \( \sqrt{2} \).
- For each \( x \), the limits for \( y \) go from \( 0 \) to \( x \).

This defines a triangular region in the \( xy \)-plane bounded by \( y = 0 \), \( y = x \), and \( x = \sqrt{2} \).

### Rewriting the bounds:
To change the order of integration, we will switch the \( dy \) and \( dx \) integrals. In the current region:
- \( x \) runs from \( y \) to \( \sqrt{2} \) (since \( x \geq y \) and \( x \leq \sqrt{2} \)).
- \( y \) runs from \( 0 \) to \( \sqrt{2} \).

Thus, the integral becomes:

\[
\int_0^{\sqrt{2}} \int_y^{\sqrt{2}} \frac{x}{\sqrt{x^2 + y^2}} \, dx \, dy
\]

This is the transformed integral with the order of integration switched.

Would you like a step-by-step breakdown of how to evaluate this integral? Here are some related questions for further exploration:

1. How does changing the order of integration simplify the evaluation of a double integral?
2. Can this integral be solved using polar coordinates instead?
3. How do we identify regions of integration in the \(xy\)-plane for more complex limits?
4. What are the common techniques for handling integrals involving square roots like \( \sqrt{x^2 + y^2} \)?
5. How would the result change if the bounds or function were modified?

**Tip:** When changing the order of integration, always sketch the region to ensure the new limits correctly describe the same area.

Change the order of integration of the double integral: \(\int_0^{\sqrt{2}} \int_0^x \frac{x}{\sqrt{x^2 + y^2}} \, dy \, dx\)

Learn how to change the order of integration for the double integral \(\int_0^{\sqrt{2}} \int_0^x \frac{x}{\sqrt{x^2 + y^2}} \, dy \, dx\). This step-by-step guide helps you understand the integration bounds and how to transform the region for easier evaluation.

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