To solve the given double integrals by reversing the order of integration, we need to analyze the limits of integration and the function inside the integrals. Let's start by rewriting the given problem.

We are given two double integrals: $\int_0^3 \int_{2x}^6 6xy \, dy \, dx + \int_3^6 \int_2^x 6xy \, dy \, dx$

Step 1: Reverse the order of integration for each integral.

First Integral:

$\int_0^3 \int_{2x}^6 6xy \, dy \, dx$ In this case, $x$ ranges from 0 to 3, and for each fixed $x$, $y$ ranges from $2x$ to 6. Reversing the order of integration requires determining the new bounds for $x$ and $y$.

• $x$ goes from 0 to 3.
• For a fixed $y$, the bounds on $x$ will depend on solving $y = 2x$, i.e., $x = \frac{y}{2}$.

Thus, we can describe the region by allowing $y$ to range from 0 to 6, and for each $y$, $x$ ranges from 0 to $\frac{y}{2}$.

Therefore, the first integral becomes: $\int_0^6 \int_0^{\frac{y}{2}} 6xy \, dx \, dy$

Second Integral:

$\int_3^6 \int_2^x 6xy \, dy \, dx$ Here, $x$ ranges from 3 to 6, and for each fixed $x$, $y$ ranges from 2 to $x$. To reverse the order of integration, we need to determine how $x$ depends on $y$.

• $y$ goes from 2 to 6.
• For each $y$, $x$ ranges from $y$ to 6.

Thus, the second integral becomes: $\int_2^6 \int_y^6 6xy \, dx \, dy$

Step 2: Combine and Solve the Reversed Integrals.

Now that we've reversed both integrals, they become: $\int_0^6 \int_0^{\frac{y}{2}} 6xy \, dx \, dy + \int_2^6 \int_y^6 6xy \, dx \, dy$

Next, we can evaluate these integrals to find the final result.

Would you like me to proceed with evaluating the integrals or provide more details about the process?

Here are five relative questions to deepen your understanding:

1. What are the geometric interpretations of reversing the order of integration?
2. How does the region of integration change when swapping the variables?
3. What are the conditions under which we can reverse the order of integration?
4. How does reversing the order affect the complexity of solving the integral?
5. Can you apply this technique to non-rectangular regions of integration?

Tip: Always visualize the region of integration when reversing the order of integration. It helps in understanding the limits and ensures the correctness of the new bounds.