∫_0^2▒(∫_0^3▒〖〖(x〗^2+2y)dy〗)dx

To solve the double integral 

\[
\int_{0}^{2} \left(\int_{0}^{3} \left(x^2 + 2y\right) dy\right) dx,
\]

we'll start by evaluating the inner integral with respect to \( y \).

### Step 1: Evaluate the inner integral

The inner integral is:

\[
\int_{0}^{3} \left(x^2 + 2y\right) dy.
\]

Here, \( x^2 \) is treated as a constant with respect to \( y \), so we can integrate each term separately:

\[
\int_{0}^{3} x^2 \, dy + \int_{0}^{3} 2y \, dy.
\]

1. **Integrate \( x^2 \) with respect to \( y \)**:
   \[
   \int x^2 \, dy = x^2 \cdot y \bigg|_{0}^{3} = x^2 \cdot 3 - x^2 \cdot 0 = 3x^2.
   \]

2. **Integrate \( 2y \) with respect to \( y \)**:
   \[
   \int 2y \, dy = y^2 \bigg|_{0}^{3} = 3^2 - 0^2 = 9.
   \]

So, the inner integral becomes:

\[
3x^2 + 9.
\]

### Step 2: Evaluate the outer integral

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

\[
\int_{0}^{2} \left(3x^2 + 9\right) dx.
\]

We can again integrate each term separately:

1. **Integrate \( 3x^2 \) with respect to \( x \)**:
   \[
   \int 3x^2 \, dx = x^3 \bigg|_{0}^{2} = 2^3 - 0^3 = 8.
   \]

2. **Integrate \( 9 \) with respect to \( x \)**:
   \[
   \int 9 \, dx = 9x \bigg|_{0}^{2} = 9(2) - 9(0) = 18.
   \]

So, the outer integral becomes:

\[
3 \cdot 8 + 18 = 24 + 18 = 42.
\]

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

\[
\boxed{42}.
\]

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the integral change if the limits of \( y \) were different?
2. What if the function inside the integral was \( x^3 + y^2 \) instead?
3. How would you solve the double integral if the function was \( e^{x+y} \)?
4. Can you apply the method of substitution to solve a double integral?
5. How do double integrals change when using polar coordinates?

**Tip:** When solving double integrals, always evaluate the inner integral first and remember that variables outside the inner integral can be treated as constants during that integration.

Learn how to evaluate the double integral ∫_0^2 ( ∫_0^3 (x^2 + 2y) dy ) dx step-by-step with detailed solutions. This problem involves the use of the Fundamental Theorem of Calculus and is suitable for advanced college-level mathematics.

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