To evaluate the double integral by changing the order of integration, we first need to understand the original limits of integration.

The given double integral is:

$\int_{0}^{3} \int_{1}^{4} yx \, dx \, dy$

The limits for $x$ range from 1 to 4, and the limits for $y$ range from 0 to 3.

Step 1: Visualizing the region of integration

The region of integration is described by the bounds:

• $x$ ranges from 1 to 4.
• $y$ ranges from 0 to 3.

So, the region $R$ is a rectangle with the following corners:

• Lower-left corner: (1, 0)
• Upper-right corner: (4, 3)

Step 2: Changing the order of integration

When we change the order of integration, the new limits for the double integral will swap the roles of $x$ and $y$. Therefore, in terms of the region $R$, we can write:

• $x$ ranges from 1 to 4.
• $y$ ranges from 0 to 3.

The new order will have $x$ as the outer integral, and $y$ as the inner one. So, the new double integral becomes:

$\int_{1}^{4} \int_{0}^{3} yx \, dy \, dx$

Step 3: Evaluating the new double integral

Now we can solve the integral:

$\int_{1}^{4} \left( \int_{0}^{3} yx \, dy \right) dx$

1. First, evaluate the inner integral: $\int_{0}^{3} yx \, dy = x \cdot \left[ \frac{y^2}{2} \right]_{0}^{3} = x \cdot \frac{9}{2} = \frac{9x}{2}$

2. Now, evaluate the outer integral: $\int_{1}^{4} \frac{9x}{2} \, dx = \frac{9}{2} \cdot \int_{1}^{4} x \, dx = \frac{9}{2} \cdot \left[ \frac{x^2}{2} \right]_{1}^{4}$

3. Calculate the result: $= \frac{9}{2} \cdot \left( \frac{16}{2} - \frac{1}{2} \right) = \frac{9}{2} \cdot \frac{15}{2} = \frac{135}{4}$

Final Answer:

$\boxed{\frac{135}{4}}$

Would you like a further breakdown or have any questions? Here are some related questions for further exploration:

1. How does changing the order of integration affect the limits?
2. What is the geometric interpretation of double integrals?
3. How can you solve integrals with different variable dependencies?
4. What happens when the region of integration is not rectangular?
5. How can you determine when to change the order of integration?

Tip: Always visualize the region of integration when changing the order of integration to ensure the new limits are correct.