Find the area of the region bounded by the curve `y^(2)=2x" and "x^(2)+y^(2)=4x`.

Explanation

To find the area of the region bounded by the curves \( y^2 = 2x \) and \( x^2 + y^2 = 4x \), we will follow these steps:

### Step 1: Identify the curves
The first curve is a parabola given by:
\[ y^2 = 2x \]
The second curve is a circle, which can be rewritten as:
\[ x^2 + y^2 - 4x = 0 \implies (x - 2)^2 + y^2 = 4 \]
This represents a circle centered at \( (2, 0) \) with a radius of \( 2 \).

### Step 2: Find the points of intersection
To find the points where the curves intersect, we substitute \( y^2 = 2x \) into the circle's equation:
\[ x^2 + (2x) - 4x = 0 \implies x^2 - 2x = 0 \implies x(x - 2) = 0 \]
Thus, \( x = 0 \) and \( x = 2 \).

Now, substituting these \( x \)-values back to find \( y \):
- For \( x = 0 \):
\[ y^2 = 2(0) \implies y = 0 \]
- For \( x = 2 \):
\[ y^2 = 2(2) \implies y = \pm 2 \]

The points of intersection are \( (0, 0) \), \( (2, 2) \), and \( (2, -2) \).

### Step 3: Set up the area integral
The area can be calculated by integrating the difference between the upper curve and the lower curve. From \( x = 0 \) to \( x = 2 \), the upper curve is the parabola \( y = \sqrt{2x} \) and the lower curve is the x-axis (i.e., \( y = 0 \)). From \( x = 2 \) to \( x = 4 \), the upper curve is the circle, which we can express as:
\[ y = \sqrt{4 - (x - 2)^2} \]
Thus, the area \( A \) can be expressed as:
\[ A = \int_0^2 \sqrt{2x} \, dx + \int_2^4 \sqrt{4 - (x - 2)^2} \, dx \]

### Step 4: Calculate the first integral
Calculating the first integral:
\[ \int_0^2 \sqrt{2x} \, dx = \int_0^2 \sqrt{2} \cdot x^{1/2} \, dx = \sqrt{2} \cdot \left[ \frac{x^{3/2}}{3/2} \right]_0^2 = \sqrt{2} \cdot \left[ \frac{2^{3/2}}{3/2} - 0 \right] = \sqrt{2} \cdot \frac{2\sqrt{2}}{3} = \frac{4}{3} \]

### Step 5: Calculate the second integral
For the second integral:
\[ \int_2^4 \sqrt{4 - (x - 2)^2} \, dx \]
This integral represents the area of a semicircle of radius \( 2 \):
\[ \text{Area} = \frac{1}{2} \cdot \pi \cdot r^2 = \frac{1}{2} \cdot \pi \cdot 2^2 = 2\pi \]

### Step 6: Combine the areas
The total area \( A \) is:
\[ A = \frac{4}{3} + 2\pi \]

### Step 7: Final area calculation
Since the area is symmetric about the x-axis, we multiply the area above the x-axis by \( 2 \):
\[ \text{Total Area} = 2 \left( \frac{4}{3} + 2\pi \right) = \frac{8}{3} + 4\pi \]

Thus, the area of the region bounded by the curves is:
\[ \boxed{\frac{8}{3} + 4\pi} \]