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To solve part (b) of this problem, we need to change the given Cartesian integral into an equivalent polar integral and evaluate it.

### Step 1: Set up the Cartesian Integral
The integral is:
\[
\int_0^1 \int_0^{y^2} 3y^3 e^{xy} \, dy \, dx
\]

### Step 2: Convert to Polar Coordinates
To convert this integral into polar coordinates, we need to express \(x\) and \(y\) in terms of \(r\) and \(\theta\):
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)

The Jacobian for converting to polar coordinates is \( dx \, dy = r \, dr \, d\theta \).

### Step 3: Determine the Limits in Polar Coordinates
1. **\(y\)-limits**: \(y\) goes from \(0\) to \(y^2\). This suggests \(y\) is bounded by a parabola, \(x = y^2\), which is harder to convert directly. So, it's better to sketch or analyze if possible.



Please solve part (b) of the problem: Convert the given Cartesian integral into an equivalent polar integral and evaluate it.

Learn how to convert a Cartesian integral to polar coordinates and evaluate it. This solution explores polar transformation, exponential functions, and double integration techniques. Suitable for Calculus II students.

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