To change the order of integration for the given integral, let's rewrite it step by step.

You mentioned the outer integral goes from $0$ to $\frac{\pi}{4}$, and the inner integral goes from $x$ to $0$. This suggests the region of integration is described by these bounds:

• $x$ is integrated from $x = 0$ to $x = \frac{\pi}{4}$,
• For each fixed $x$, $y$ is integrated from $y = x$ to $y = 0$.

Now, let's change the order of integration:

1. Describe the Region

The region of integration is the triangle in the $xy$-plane bounded by the lines:

• $y = 0$ (horizontal axis),
• $y = x$,
• $x = \frac{\pi}{4}$.

2. New Limits for $x$ and $y$

When changing the order of integration, we need to rethink the limits:

• $y$ will now range from $0$ to $\frac{\pi}{4}$,
• For each fixed $y$, $x$ will range from $y$ to $\frac{\pi}{4}$.

3. Rewriting the Integral

The new limits for $x$ and $y$ are:

• Outer integral (with respect to $y$): from $0$ to $\frac{\pi}{4}$,
• Inner integral (with respect to $x$): from $y$ to $\frac{\pi}{4}$.

Thus, the new integral becomes:

$\int_{0}^{\frac{\pi}{4}} \int_{y}^{\frac{\pi}{4}} f(x, y) \, dx \, dy$

Would you like more details about the steps, or do you have any questions?

5 Questions for Further Exploration:

1. What happens to the limits if the region of integration is not a triangle?
2. How would the limits change if the outer bounds were from $0$ to $\pi/2$?
3. Can we apply Fubini's theorem in all cases of changing order of integration?
4. How does the nature of $f(x, y)$ affect the integration strategy?
5. What if $f(x, y)$ is separable as $f(x) \cdot g(y)$?

Tip:

Always sketch the region of integration when changing the order of integration. It helps visualize the correct bounds.