The given integral in the image is written as:

$\int_0^7 \sin x \, \bigg( \int_0^7 f(x,y) \, dy \bigg) dx$

If you want to reverse the order of integration $dy$ and $dx$, you need to consider the bounds and ensure the integral remains mathematically valid. This generally assumes $f(x, y)$ is well-behaved and integrable over the specified region.

Rewriting the integral with reversed order:

The integral becomes:

$\int_0^7 \bigg( \int_0^7 \sin x \cdot f(x, y) \, dx \bigg) dy$

Explanation:

• The order of integration is reversed by bringing the $dx$-dependent part inside the $dy$ integral.
• Here, $f(x, y)$ remains the same but is now multiplied by $\sin x$ inside the $dx$ integration.

Let me know if you'd like detailed steps or further questions! 😊

5 Related Questions:

1. What conditions must $f(x, y)$ satisfy to switch the order of integration?
2. How would the result differ if the bounds of integration were not constants but dependent on $x$ or $y$?
3. Can this approach be applied to triple integrals? How?
4. How does reversing the order affect applications like finding areas or volumes?
5. What are the physical implications of switching integration order in problems like flux or mass distributions?

One Tip:

When reversing integration order, always verify the bounds carefully, especially for regions where one variable depends on the other!