The image presents a double integral transformation, where the order of integration is being swapped from $\int_{-5}^{5} \int_{x^2}^{25} f(x,y) \, dy \, dx$ to a different set of limits for integration in terms of $dx \, dy$.

Let's break down the problem and solve it:

Initial Integral Setup:

$\int_{-5}^{5} \int_{x^2}^{25} f(x,y) \, dy \, dx$ This is the double integral in the order $dy \, dx$, with the limits:

• For $x$, the limits are $x = -5$ to $x = 5$.
• For $y$, the limits are $y = x^2$ to $y = 25$.

Changing the Order of Integration:

1. Interpret the Region of Integration:

   • The variable $y$ ranges from $x^2$ to 25, which means $y \geq x^2$, so $y$ starts at the parabola $y = x^2$ and goes up to the constant value 25.
   • The variable $x$ ranges from $-5$ to $5$.

   We need to understand this region in the $xy$-plane:

   • For $y$, the lowest point occurs at $y = 0$ (the vertex of the parabola $y = x^2$ when $x = 0$).
   • For $y = 25$, $x^2 \leq 25$, so $x$ ranges from $-\sqrt{25} = -5$ to $\sqrt{25} = 5$.

2. New Limits for $dx \, dy$ Order:

   • The limits for $y$ are now from $0$ to $25$.
   • For each fixed $y$, $x$ ranges from $-\sqrt{y}$ to $\sqrt{y}$ (since $y = x^2$).

Transformed Integral:

Thus, the new integral with the order of integration swapped becomes: $\int_{0}^{25} \int_{-\sqrt{y}}^{\sqrt{y}} f(x,y) \, dx \, dy$

This is the correct transformation of the original integral.

Would you like more detailed explanations or have any questions on how the integration limits were determined?

Here are five follow-up questions related to this concept:

1. How do we change the limits of integration for more complex regions?
2. What types of functions benefit from swapping the order of integration?
3. How does the shape of the region affect the integration process?
4. Could we use polar coordinates to simplify a similar problem?
5. Why is it important to graph the region when switching the order of integration?

Tip: Always sketch the region of integration when switching the order, as it helps to visualize the boundaries clearly.