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To set up a definite integral that represents the area of the shaded region, we need to identify the boundaries and limits in terms of \( y \), as indicated by the presence of \( dy \) in the integral.

The shaded region is bounded by:
- The curve \( g(y) = y^3 \).
- The vertical line \( y = 2 \), which is the upper limit.
- The vertical line \( y = 0 \), which is the lower limit.

### Steps to Set Up the Integral

1. **Express the Curve in Terms of \( x \):** Since \( x = y^3 \), we can use this expression for \( x \) as a function of \( y \) in the integral.

2. **Define the Limits for \( y \):** The region extends from \( y = 0 \) to \( y = 2 \).

3. **Formulate the Integral:** The area can be given by the integral of \( x \) with respect to \( y \), from \( y = 0 \) to \( y = 2 \).

Thus, the integral for the area is:
\[
\int_{0}^{2} y^3 \, dy
\]

This setup represents the area of the region bounded by \( g(y) = y^3 \) from \( y = 0 \) to \( y = 2 \).

Would you like more details on how to interpret the bounds or on solving this integral?

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### Related Questions:
1. How do you evaluate a definite integral for a function?
2. What does it mean when an integral represents an area under a curve?
3. How would the integral change if the bounds for \( y \) were different?
4. What are the steps to find the area between two curves?
5. How can you convert between \( x \)-bound and \( y \)-bound integrals?

### Tip:
When setting up an integral for area, always carefully examine the limits of integration and whether you need to express everything in terms of \( x \) or \( y \).

Write a definite integral that yields the area of the region bounded by the curve g(y) = y^3 and the line y = 2.

Learn how to set up a definite integral to find the area under the curve g(y) = y^3 from y = 0 to y = 2. This problem focuses on integral calculus and involves the fundamental theorem of calculus. Suitable for students in Grades 11-12.

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