The area of the region bounded by the circle `x^(2)+y^(2)=1` and the line `x+y=1` is :

Options

A. `((pi)/(2)-(1)/(2))` sq. units
B. `(pi)/(4)` sq. units
C. `(1)/(2)` sq. units
D. None of these

Answer

To find the area of the region bounded by the circle \(x^2 + y^2 = 1\) and the line \(x + y = 1\), we will follow these steps:

### Step 1: Identify the points of intersection
We need to find the points where the circle and the line intersect. To do this, we can substitute \(y\) from the line equation into the circle equation.

The line equation is:
\[ y = 1 - x \]

Substituting this into the circle equation:
\[ x^2 + (1 - x)^2 = 1 \]
Expanding this:
\[ x^2 + (1 - 2x + x^2) = 1 \]
\[ 2x^2 - 2x + 1 = 1 \]
\[ 2x^2 - 2x = 0 \]
Factoring out \(2x\):
\[ 2x(x - 1) = 0 \]
Thus, \(x = 0\) or \(x = 1\).

Now substituting these \(x\) values back into the line equation to find corresponding \(y\) values:
- For \(x = 0\), \(y = 1\) (point (0, 1))
- For \(x = 1\), \(y = 0\) (point (1, 0))

### Step 2: Set up the integrals
We will calculate the area between the curve of the circle and the line from \(x = 0\) to \(x = 1\).

The area under the circle from \(x = 0\) to \(x = 1\) is given by:
\[ A_{\text{circle}} = \int_0^1 \sqrt{1 - x^2} \, dx \]

The area under the line from \(x = 0\) to \(x = 1\) is given by:
\[ A_{\text{line}} = \int_0^1 (1 - x) \, dx \]

### Step 3: Calculate the area under the circle
To compute \(A_{\text{circle}}\):
\[ A_{\text{circle}} = \int_0^1 \sqrt{1 - x^2} \, dx \]
Using the formula for the area of a quarter circle:
\[ A_{\text{circle}} = \frac{\pi}{4} \]

### Step 4: Calculate the area under the line
To compute \(A_{\text{line}}\):
\[ A_{\text{line}} = \int_0^1 (1 - x) \, dx \]
Calculating this integral:
\[ A_{\text{line}} = \left[ x - \frac{x^2}{2} \right]_0^1 = \left( 1 - \frac{1}{2} \right) - (0 - 0) = \frac{1}{2} \]

### Step 5: Calculate the area of the bounded region
The area of the region bounded by the circle and the line is:
\[ A = A_{\text{circle}} - A_{\text{line}} = \frac{\pi}{4} - \frac{1}{2} \]

### Final Answer
Thus, the area of the region bounded by the circle \(x^2 + y^2 = 1\) and the line \(x + y = 1\) is:
\[ \frac{\pi}{4} - \frac{1}{2} \]