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.