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The problem asks us to:

1. **Sketch the region of integration**.
2. **Reverse the order of integration**.
3. **Evaluate the integral**.

Given the double integral:

\[
\int_0^{\sqrt{3}} \int_{\sqrt{x}}^{\sqrt{3}} 3y \, dy \, dx
\]

Let's go through these steps:

### 1. Sketching the Region of Integration
The integral has the outer limits for \( x \) from \( 0 \) to \( \sqrt{3} \), and the inner limits for \( y \) from \( \sqrt{x} \) to \( \sqrt{3} \).

The bounds describe the region:

- \( x \) ranges from \( 0 \) to \( \sqrt{3} \).
- \( y \) ranges from \( \sqrt{x} \) to \( \sqrt{3} \).

The curve \( y = \sqrt{x} \) defines the lower boundary for \( y \), and the line \( y = \sqrt{3} \) is the upper boundary.

### 2. Reversing the Order of Integration
To reverse the order of integration, we need to describe \( x \) in terms of \( y \).

- From \( y = \sqrt{x} \), we get \( x = y^2 \).
- \( x \) ranges from \( 0 \) to \( \sqrt{3} \), and \( y \) ranges from \( 0 \) to \( \sqrt{3} \).

The region can be rewritten with \( y \) ranging from \( 0 \) to \( \sqrt{3} \) and \( x \) ranging from \( 0 \) to \( y^2 \). Thus, the reversed integral becomes:

\[
\int_0^{\sqrt{3}} \int_0^{y^2} 3y \, dx \, dy
\]

### 3. Evaluating the Integral
Now, let's evaluate the reversed integral:

\[
\int_0^{\sqrt{3}} \int_0^{y^2} 3y \, dx \, dy = \int_0^{\sqrt{3}} 3y \left[x\right]_0^{y^2} \, dy = \int_0^{\sqrt{3}} 3y \cdot y^2 \, dy = \int_0^{\sqrt{3}} 3y^3 \, dy
\]

\[
= 3 \int_0^{\sqrt{3}} y^3 \, dy = 3 \left[\frac{y^4}{4}\right]_0^{\sqrt{3}} = 3 \cdot \frac{(\sqrt{3})^4}{4} = 3 \cdot \frac{9}{4} = \frac{27}{4}
\]

Therefore, the value of the integral is:

\[
\boxed{\frac{27}{4}}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How can you determine the region of integration when given different integral limits?
2. What are some common methods to reverse the order of integration?
3. How do changes in bounds affect the area of the integration region?
4. What are some applications of double integrals in real-world problems?
5. How can changing variables (like using polar coordinates) simplify certain double integrals?

**Tip:** Always sketch the region of integration before reversing the order; it helps in setting up the correct bounds for the new order of integration.

Sketch the region of integration, reverse the order of integration, and evaluate the integral: ∫∫ (from y = √x to y = √3) (from x = 0 to x = √3) 3y dy dx.

Learn how to sketch the region of integration, reverse the order of integration, and evaluate a double integral with step-by-step guidance. This problem covers reversing limits and calculating the area of the integration region, suitable for college-level calculus.

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