Find the area bounded by the curve `y=2 cosx` and the X-axis from x = 0 to `x=2pi`.

Explanation

To find the area bounded by the curve \( y = 2 \cos x \) and the x-axis from \( x = 0 \) to \( x = 2\pi \), we will follow these steps:

### Step 1: Understand the Function and its Behavior
The function \( y = 2 \cos x \) oscillates between -2 and 2. We need to find the points where this function intersects the x-axis (where \( y = 0 \)) within the interval \( [0, 2\pi] \).

### Step 2: Find the Points of Intersection with the X-axis
Set \( 2 \cos x = 0 \):
\[ \cos x = 0 \]
The solutions in the interval \( [0, 2\pi] \) are:
\[ x = \frac{\pi}{2}, \quad x = \frac{3\pi}{2} \]

### Step 3: Set Up the Integral for Area Calculation
The area can be calculated by integrating the function from \( 0 \) to \( 2\pi \) but we need to consider the intervals where the function is above and below the x-axis.

1. From \( 0 \) to \( \frac{\pi}{2} \), \( y = 2 \cos x \) is positive.
2. From \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \), \( y = 2 \cos x \) is negative.
3. From \( \frac{3\pi}{2} \) to \( 2\pi \), \( y = 2 \cos x \) is positive again.

Thus, the area \( A \) can be expressed as:
\[ A = \int_{0}^{\frac{\pi}{2}} 2 \cos x \, dx - \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} 2 \cos x \, dx + \int_{\frac{3\pi}{2}}^{2\pi} 2 \cos x \, dx \]

### Step 4: Calculate Each Integral
1. **First Integral**:
\[ \int_{0}^{\frac{\pi}{2}} 2 \cos x \, dx = 2 \left[ \sin x \right]_{0}^{\frac{\pi}{2}} = 2 (\sin(\frac{\pi}{2}) - \sin(0)) = 2(1 - 0) = 2 \]

2. **Second Integral**:
\[ \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} 2 \cos x \, dx = 2 \left[ \sin x \right]_{\frac{\pi}{2}}^{\frac{3\pi}{2}} = 2 (\sin(\frac{3\pi}{2}) - \sin(\frac{\pi}{2})) = 2(-1 - 1) = -4 \]
Since this area is below the x-axis, we take the absolute value:
\[ \text{Area} = 4 \]

3. **Third Integral**:
\[ \int_{\frac{3\pi}{2}}^{2\pi} 2 \cos x \, dx = 2 \left[ \sin x \right]_{\frac{3\pi}{2}}^{2\pi} = 2 (\sin(2\pi) - \sin(\frac{3\pi}{2})) = 2(0 - (-1)) = 2 \]

### Step 5: Combine the Areas
Now, we sum the areas calculated:
\[ A = 2 + 4 + 2 = 8 \]

### Final Answer
The area bounded by the curve \( y = 2 \cos x \) and the x-axis from \( x = 0 \) to \( x = 2\pi \) is \( 8 \) square units.

---