change order of integral outer integral 0 to pi/4 inner x to 0. 

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.

change order of integral outer integral 0 to pi/4 inner x to 0

Learn how to change the order of integration for a double integral with limits 0 to pi/4 for the outer integral and x to 0 for the inner integral. This detailed step-by-step solution explains how to identify the region of integration and apply new limits. Ideal for students in Calculus II or Multivariable Calculus courses.

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