The ratio of the areas of two regions of the curve `C_(1)-=4x^(2)+pi^(2)y^(2)=4pi^(2)` divided by the curve `C_(2)-=y=-(sgn(x-(pi)/(2)))cosx` (where sgn (x) = signum (x)) is

Options

A. `(pi^(2)-2)/(pi^(2)-2sqrt2)`
B. `(pi^(2)+2)/(pi^(2)-2sqrt2)`
C. `(pi^(2)+6)/(pi^(2)+3sqrt2)`
D. `(pi^(2)-1)/(pi^(2)-sqrt2)`

Explanation

To solve the problem of finding the ratio of the areas of two regions defined by the curves \( C_1 \) and \( C_2 \), we will follow these steps:

### Step 1: Analyze the curves
The first curve \( C_1 \) is given by the equation:
\[ 4x^2 + \pi^2 y^2 = 4\pi^2 \]
This can be rewritten in standard form as:
\[ \frac{x^2}{\pi^2} + \frac{y^2}{4} = 1 \]
This represents an ellipse centered at the origin with semi-major axis \( 2 \) along the \( y \)-axis and semi-minor axis \( \pi \) along the \( x \)-axis.

The second curve \( C_2 \) is given by:
\[ y = -\text{sgn}(x - \frac{\pi}{2}) \cos x \]
This means:
- For \( x > \frac{\pi}{2} \), \( y = -\cos x \)
- For \( x = \frac{\pi}{2} \), \( y = 0 \)
- For \( x < \frac{\pi}{2} \), \( y = \cos x \)

### Step 2: Calculate the area of the region bounded by \( C_1 \)
The area \( A_1 \) of the ellipse can be calculated using the formula for the area of an ellipse:
\[ \text{Area} = \pi \times a \times b \]
where \( a \) and \( b \) are the semi-major and semi-minor axes respectively. Here, \( a = 2 \) and \( b = \pi \):
\[ A_1 = \pi \times 2 \times \pi = 2\pi^2 \]

### Step 3: Calculate the area of the region bounded by \( C_2 \)
To find the area \( A_2 \), we need to consider the integral of the function \( y = -\cos x \) from \( x = \frac{\pi}{2} \) to \( x = \frac{3\pi}{2} \) (the region where \( y \) is below the x-axis):
\[ A_2 = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} -\cos x \, dx \]
Calculating this integral:
\[ A_2 = -\left[ \sin x \right]_{\frac{\pi}{2}}^{\frac{3\pi}{2}} = -(\sin(\frac{3\pi}{2}) - \sin(\frac{\pi}{2})) = -(-1 - 1) = 2 \]

### Step 4: Find the ratio of the areas
Now that we have both areas, we can find the ratio:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{2\pi^2}{2} = \pi^2 \]

### Final Answer
The ratio of the areas of the two regions is:
\[ \text{Ratio} = \pi^2 \]