Find the area of that region bounded by the curve `y="cos"x, ` X-axis, `x=0` and `x=pi`.

Answer

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

### Step 1: Understand the Region
We need to visualize the region bounded by the curve \( y = \cos x \), the x-axis, and the vertical lines \( x = 0 \) and \( x = \pi \). The curve \( y = \cos x \) starts at \( (0, 1) \) and decreases to \( (0, 0) \) at \( x = \pi \).

### Step 2: Identify the Points of Intersection
The curve intersects the x-axis at points where \( y = 0 \). The cosine function equals zero at \( x = \frac{\pi}{2} \). Thus, the area we are interested in is from \( x = 0 \) to \( x = \pi \).

### Step 3: Set Up the Integral
The area \( A \) under the curve from \( x = 0 \) to \( x = \pi \) can be calculated using the integral:
\[ A = \int_{0}^{\pi} \cos x \, dx \]

### Step 4: Evaluate the Integral
To evaluate the integral, we find the antiderivative of \( \cos x \):
\[ \int \cos x \, dx = \sin x \]
Now, we evaluate this from \( 0 \) to \( \pi \):
\[ A = \left[ \sin x \right]_{0}^{\pi} = \sin(\pi) - \sin(0) = 0 - 0 = 0 \]

### Step 5: Calculate the Area Above the x-axis
Since the curve is above the x-axis from \( 0 \) to \( \frac{\pi}{2} \) and below the x-axis from \( \frac{\pi}{2} \) to \( \pi \), we need to split the integral:
\[ A = \int_{0}^{\frac{\pi}{2}} \cos x \, dx + \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx \]

### Step 6: Evaluate Each Integral
1. For the first integral:
\[ \int_{0}^{\frac{\pi}{2}} \cos x \, dx = \left[ \sin x \right]_{0}^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

2. For the second integral:
\[ \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx = -\left[ \sin x \right]_{\frac{\pi}{2}}^{\pi} = -(\sin(\pi) - \sin\left(\frac{\pi}{2}\right)) = -\left(0 - 1\right) = 1 \]

### Step 7: Combine the Areas
Now, we add the areas from both integrals:
\[ A = 1 + 1 = 2 \]

### Final Answer
Thus, the area of the region bounded by the curve \( y = \cos x \), the x-axis, \( x = 0 \), and \( x = \pi \) is:
\[ \text{Area} = 2 \text{ square units} \]

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