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We are tasked with evaluating the double integral:

\[
\iint_R 6xy e^{x^2 y} \, dA
\]

where the region \( R = [0, 1] \times [0, 2] \).

### Steps to solve:

1. **Set up the double integral**:
   The double integral can be written as:
   \[
   \int_0^1 \left( \int_0^2 6xy e^{x^2 y} \, dy \right) dx
   \]
   
2. **Solve the inner integral**:
   We first focus on the inner integral with respect to \( y \):
   \[
   \int_0^2 6xy e^{x^2 y} \, dy
   \]
   We can use substitution to simplify this:
   Let \( u = x^2 y \), hence \( du = x^2 dy \), or equivalently \( dy = \frac{du}{x^2} \).

   Now, rewrite the limits for \( y = 0 \) and \( y = 2 \):
   - When \( y = 0 \), \( u = 0 \).
   - When \( y = 2 \), \( u = 2x^2 \).

   So the integral becomes:
   \[
   \frac{6}{x} \int_0^{2x^2} u e^u \, du
   \]
   The integral of \( u e^u \) is \( (u - 1) e^u \), so evaluating from 0 to \( 2x^2 \) gives:
   \[
   \frac{6}{x} \left[ (2x^2 - 1) e^{2x^2} - (-1)e^0 \right]
   \]
   Simplifying:
   \[
   \frac{6}{x} \left[ (2x^2 - 1) e^{2x^2} + 1 \right]
   \]

3. **Solve the outer integral**:
   Now we integrate with respect to \( x \) from 0 to 1:
   \[
   \int_0^1 \frac{6}{x} \left[ (2x^2 - 1) e^{2x^2} + 1 \right] dx
   \]
   This integral is complex but can be handled using a combination of integration techniques (by parts and substitution) or calculated numerically.

Let me know if you'd like a detailed numeric computation or further steps.

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Would you like more details on any specific step?

Here are 5 related questions:
1. What is the importance of substituting variables in integrals?
2. Can you generalize this method to any exponential function of polynomials?
3. How does changing the region of integration affect the result?
4. What is the best method for solving complex integrals numerically?
5. How can the Jacobian be used in integrals over more complex regions?

**Tip**: In double integrals, it’s helpful to compute the inner integral first, simplifying the remaining outer integral step.

Calculate the double integral \(\iint_R 6xy e^{x^2 y} \, dA\) over the region \( R = [0,1] \times [0,2] \).

Learn how to evaluate the double integral \(\iint_R 6xy e^{x^2 y} \, dA\) over the region \( R = [0,1] \times [0,2] \) using substitution and exponential integration. Suitable for advanced calculus students.

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