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To rewrite the iterated integral \(\int_0^7 \int_{3x/7}^3 f(x,y) \, dy \, dx\) in the opposite order, we need to analyze the limits of integration and switch the order of integration from \(dy \, dx\) to \(dx \, dy\).

### Step 1: Understand the region of integration
- The outer integral is from \(x = 0\) to \(x = 7\).
- The inner integral has the limits \(y = \frac{3x}{7}\) to \(y = 3\).

We need to express this region in terms of \(y\) first (outer integral) and \(x\) second (inner integral).

### Step 2: Find the limits of \(x\) in terms of \(y\)
1. From the given limits \(y = \frac{3x}{7}\), solve for \(x\) in terms of \(y\):
   \[
   y = \frac{3x}{7} \quad \Rightarrow \quad x = \frac{7y}{3}.
   \]
   
2. When \(y = 3\), the upper bound for \(x\) is \(7\), and when \(y = 0\), the lower bound for \(x\) is \(0\).

### Step 3: Set up the new limits
- \(y\) will range from 0 to 3 (outer integral).
- For each \(y\), \(x\) will range from \(\frac{7y}{3}\) to 7.

### Step 4: Write the new integral
The integral in the opposite order is:
\[
\int_0^3 \int_{\frac{7y}{3}}^7 f(x, y) \, dx \, dy.
\]

Would you like more details or further clarification? 

### Related Questions:
1. What are the conditions for switching the order of integration in iterated integrals?
2. How do you graph the region of integration for double integrals?
3. Can you always switch the order of integration for any given region?
4. What happens if the limits are dependent in a different way (not linear)?
5. How does Fubini's theorem relate to switching the order of integration?

### Tip:
When switching the order of integration, always visualize the region of integration carefully and rewrite the boundaries accordingly.

Express the iterated integral in the opposite order: ∫₀⁷ ∫₃ₓ/₇³ f(x, y) dy dx

Learn how to switch the order of integration for the iterated integral ∫₀⁷ ∫₃ₓ/₇³ f(x, y) dy dx. This problem demonstrates the use of Fubini's Theorem and is suitable for students in Calculus II or Multivariable Calculus.

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