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The original integral given is:

\[
\int_0^7 \int_{3x/7}^{3} f(x, y) \, dy \, dx
\]

To reverse the order of integration (from \(dy \, dx\) to \(dx \, dy\)), we need to:

1. Analyze the bounds:
   - The outer integral (with respect to \(x\)) has limits from \(0\) to \(7\).
   - The inner integral (with respect to \(y\)) has limits from \(\frac{3x}{7}\) to \(3\).

2. Determine how these limits relate to the variables.
   - The lower limit of \(y\) depends on \(x\) (\(\frac{3x}{7}\)), while the upper limit is a constant (3).
   - Rewrite these relationships for \(x\) in terms of \(y\):
     \[
     \frac{3x}{7} = y \implies x = \frac{7y}{3}
     \]
   So for \(y\), \(x\) varies between \(0\) and \(\frac{7y}{3}\).

3. Set the new bounds for the reversed order of integration:
   - \(y\) ranges from \(0\) to \(3\).
   - For each \(y\), \(x\) ranges from \(0\) to \(\frac{7y}{3}\).

Thus, the iterated integral in the opposite order is:

\[
\int_0^3 \int_0^{\frac{7y}{3}} f(x, y) \, dx \, dy
\]

This matches the fourth option (the one that is circled):

\[
\int_0^3 \int_0^{\frac{7y}{3}} f(x, y) \, dx \, dy
\]

Would you like more details on how the bounds were derived, or do you have any questions?

Here are 5 related questions for practice:
1. How do you change the order of integration for a general iterated integral?
2. What are the steps to finding new bounds when reversing the order of integration?
3. Why is it necessary to express \(x\) as a function of \(y\) when reversing the integration order?
4. Can you change the order of integration if the function has discontinuities within the bounds?
5. How do you verify that the new bounds are correct after changing the order?

**Tip:** When reversing the order of integration, it's crucial to sketch the region of integration to visualize how the bounds transform.

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

Learn how to reverse the order of integration for the double integral ∫₀⁷ ∫₃ₓ/₇³ f(x, y) dy dx. This solution involves analyzing the bounds and using Fubini's Theorem. Suitable for college-level calculus students.

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