To solve the double integral 

\[
\iint_{R} (3x^2 y - 2xy) \, dx \, dy,
\]

where \( R = [1, 2] \times [1, 2] \), we'll integrate over the region \( R \) which is a rectangle in the \( xy \)-plane.

### Step 1: Set up the double integral

Given the region \( R = [1, 2] \times [1, 2] \), the double integral is:

\[
\int_{1}^{2} \left( \int_{1}^{2} (3x^2 y - 2xy) \, dx \right) dy.
\]

### Step 2: Evaluate the inner integral

Let's first evaluate the inner integral with respect to \( x \):

\[
\int_{1}^{2} (3x^2 y - 2xy) \, dx.
\]

We'll integrate each term separately:

1. **Integrate \( 3x^2 y \) with respect to \( x \)**:
   \[
   \int 3x^2 y \, dx = y \cdot \int 3x^2 \, dx = y \cdot x^3 \bigg|_{1}^{2} = y \cdot \left(8 - 1\right) = 7y.
   \]

2. **Integrate \( -2xy \) with respect to \( x \)**:
   \[
   \int -2xy \, dx = -2y \cdot \int x \, dx = -2y \cdot \frac{x^2}{2} \bigg|_{1}^{2} = -y \cdot \left(4 - 1\right) = -3y.
   \]

So, the inner integral becomes:

\[
7y - 3y = 4y.
\]

### Step 3: Evaluate the outer integral

Now, we evaluate the outer integral with respect to \( y \):

\[
\int_{1}^{2} 4y \, dy.
\]

Integrating \( 4y \) with respect to \( y \):

\[
4 \cdot \frac{y^2}{2} \bigg|_{1}^{2} = 2y^2 \bigg|_{1}^{2} = 2(4) - 2(1) = 8 - 2 = 6.
\]

### Final Answer:
The value of the double integral is:

\[
\boxed{6}.
\]

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How does the integral change if the region \( R \) is different, such as \( R = [0, 1] \times [1, 3] \)?
2. What happens if the integrand is \( x^2 - y^2 \) instead?
3. How would you solve this integral if \( R \) was a triangular region instead of a rectangle?
4. What if the function inside the integral was \( e^{xy} \)?
5. How does the order of integration affect the result in double integrals?

**Tip:** When solving double integrals over rectangular regions, always evaluate one integral at a time, treating the other variable as a constant.

Learn how to solve the double integral ∬(3x^2 y - 2xy) dxdy over the rectangular region R = [1;2]x[1;2]. This problem involves integrating a polynomial function over a specified region in the xy-plane. Suitable for undergraduate students studying calculus and integration.

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