The problem is to find the area of a region defined by the inequalities:

1. $xy \geq 4$
2. $y \leq x$
3. $27y \geq 4x^2$

These inequalities define a bounded region in the coordinate plane.

Solution Steps

To calculate the area, we use double integrals with appropriate limits derived from the inequalities:

1. Setting up the inequalities:

   • $xy = 4$ represents a hyperbola.
   • $y = x$ is the line $y = x$.
   • $27y = 4x^2$ is a parabola with $y = \frac{4x^2}{27}$.

2. Intersection points: Solving the intersections will help us define the limits of integration for each integral.

3. Integration setup: The solution uses different limits and integrands, shown in the provided image. It splits the region into two main integrals, as seen in the setup:

   • The first integral is set up as: $\int_0^2 \int_0^3 (4 - y^2) \, dx \, dy$
   • Then, the second integral: $\int_0^4 \int_{\frac{\sqrt{y}}{3}}^{y/2} (4x + 2) \, dx \, dy$

These integrals can now be solved individually to obtain the area of the specified region.

Would you like me to proceed with detailed integration steps for these integrals?

Further Questions

1. What are the exact steps to find the intersection points between the curves?
2. How do we evaluate each integral individually?
3. How does changing the order of integration affect the solution?
4. Can the area be calculated using a polar coordinate transformation?
5. How can the inequalities be interpreted geometrically to understand the bounded region?

Tip

For problems involving multiple inequalities, sketching the region defined by each curve can simplify the process of setting integration limits.