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.