Find the area of the semi-portion of the circle `x ^(2) + y ^(2) =4.`

Answer

To find the area of the semi-portion of the circle given by the equation \( x^2 + y^2 = 4 \), we can follow these steps:

### Step 1: Identify the Circle's Properties
The equation \( x^2 + y^2 = 4 \) represents a circle with:
- Center at the origin (0, 0)
- Radius \( r = \sqrt{4} = 2 \)

### Step 2: Calculate the Area of the Full Circle
The area \( A \) of a full circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (2)^2 = 4\pi \]

### Step 3: Calculate the Area of the Semi-Circle
Since we need the area of the semi-portion (half of the circle), we divide the area of the full circle by 2:
\[ \text{Area of semi-circle} = \frac{1}{2} A = \frac{1}{2} (4\pi) = 2\pi \]

### Final Answer
The area of the semi-portion of the circle \( x^2 + y^2 = 4 \) is:
\[ \boxed{2\pi} \]

---