Key Concepts and Formulas

• Area of an Ellipse: The area of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is given by $A_{ellipse} = \pi ab$.
• Area of a Rhombus: The area of a rhombus (or diamond) formed by $|x|/p + |y|/q = 1$ is $2pq$.

Step-by-Step Solution

Step 1: Identify the Equations and Parameters

We are given the equation of the ellipse as $\frac{x^2}{4} + \frac{y^2}{9} = 1$. From this equation, we can identify the semi-major and semi-minor axes as $a = \sqrt{4} = 2$ and $b = \sqrt{9} = 3$.

We are also given the equation of the region shaped like a diamond as $\frac{|x|}{2} + \frac{|y|}{3} = 1$. This is a rhombus centered at the origin. From this equation, we can identify $p = 2$ and $q = 3$.

Step 2: Calculate the Area of the Ellipse

Using the formula for the area of an ellipse, $A_{ellipse} = \pi ab$, we substitute $a = 2$ and $b = 3$ to get: $A_{ellipse} = \pi (2)(3) = 6\pi$

Step 3: Calculate the Area of the Rhombus

The equation $\frac{|x|}{2} + \frac{|y|}{3} = 1$ represents a rhombus. The area of the rhombus is given by $2pq$. Substituting $p = 2$ and $q = 3$ we get: $A_{rhombus} = 2(2)(3) = 12$

Step 4: Calculate the Area of the Region Outside the Rhombus and Inside the Ellipse

The area we seek is the area of the ellipse minus the area of the rhombus. $A = A_{ellipse} - A_{rhombus} = 6\pi - 12 = 6(\pi - 2)$

Common Mistakes & Tips

• Be careful to correctly identify the semi-major and semi-minor axes of the ellipse. It's easy to mix them up if the equation isn't in standard form.
• The area of the region $|x|/p + |y|/q = 1$ is a standard result. Memorizing it saves time. It can be easily derived by recognizing that the region is a rhombus composed of 4 congruent triangles, each with area $pq/2$.
• Always visualize the regions to confirm that you're subtracting the correct areas.

Summary

We first identified the parameters of the ellipse and the rhombus from their equations. Then, we calculated the area of the ellipse and the area of the rhombus. Finally, we subtracted the area of the rhombus from the area of the ellipse to find the area of the region outside the rhombus and inside the ellipse. The final area is $6(\pi - 2)$ square units.

Final Answer

The final answer is \boxed{6(\pi - 2)}, which corresponds to option (C).