Area of the region bounded by the curve `{(x,y) : (x^(2))/(a^(2)) + (y^(2))/(b^(2)) le 1 le "" (x)/(a) + (y)/(b)}` is

Options

A. `((pi)/(4)-(1)/(2))ab`
B. `((pi)/(4)+(1)/(2))ab`
C. `((pi)/(4)-(1)/(3))ab`
D. `(pi)/(4) ab`

Answer

To find the area of the region bounded by the curves given by the equations \( \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \) and \( \frac{x}{a} + \frac{y}{b} \geq 1 \), we will follow these steps:

### Step 1: Understand the Curves
The first equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) represents an ellipse centered at the origin with semi-major axis \( b \) along the y-axis and semi-minor axis \( a \) along the x-axis.

The second equation \( \frac{x}{a} + \frac{y}{b} = 1 \) represents a straight line that intersects the axes at points \( (a, 0) \) and \( (0, b) \).

### Step 2: Identify the Region
We need to find the area of the region that lies inside the ellipse and above the line. The line divides the plane into two regions, and we are interested in the area above the line and inside the ellipse.

### Step 3: Find Intersection Points
To find the intersection points of the ellipse and the line, we can substitute \( y \) from the line equation into the ellipse equation.

From the line equation:
\[ y = b - \frac{b}{a}x \]

Substituting into the ellipse equation:
\[ \frac{x^2}{a^2} + \frac{(b - \frac{b}{a}x)^2}{b^2} = 1 \]

Expanding and simplifying this will give us the x-coordinates of the intersection points.

### Step 4: Set Up the Integral
The area we want to find can be expressed as:
\[ \text{Area} = \int_{0}^{a} \left( y_{\text{ellipse}} - y_{\text{line}} \right) \, dx \]

Where \( y_{\text{ellipse}} \) is derived from the ellipse equation and \( y_{\text{line}} \) is derived from the line equation.

### Step 5: Calculate \( y_{\text{ellipse}} \)
From the ellipse equation:
\[ y = b \sqrt{1 - \frac{x^2}{a^2}} \]

### Step 6: Calculate \( y_{\text{line}} \)
From the line equation:
\[ y = b - \frac{b}{a}x \]

### Step 7: Set Up the Integral
Now, we can set up the integral:
\[ \text{Area} = \int_{0}^{a} \left( b \sqrt{1 - \frac{x^2}{a^2}} - \left(b - \frac{b}{a}x\right) \right) dx \]

### Step 8: Simplify the Integral
This simplifies to:
\[ \text{Area} = \int_{0}^{a} \left( b \sqrt{1 - \frac{x^2}{a^2}} - b + \frac{b}{a}x \right) dx \]

### Step 9: Evaluate the Integral
Now we can evaluate the integral term by term:
1. The integral of \( b \sqrt{1 - \frac{x^2}{a^2}} \)
2. The integral of \( -b \)
3. The integral of \( \frac{b}{a}x \)

### Step 10: Combine Results
After evaluating these integrals from \( 0 \) to \( a \), we combine the results to find the total area.

### Final Result
The area of the region bounded by the curves is:
\[ \text{Area} = \frac{ab}{2} \left( \frac{\pi}{2} - 1 \right) \]