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

Options

A. 2 sq units
B. 4 sq units
C. 3 sq units
D. 1 sq unit

Answer

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

### Step 1: Identify the area to be calculated
The area we are looking for is bounded by:
- The curve \( y = \cos x \)
- The X-axis (where \( y = 0 \))
- The vertical lines \( x = 0 \) and \( x = \pi \)

### Step 2: Sketch the graph
Sketching the graph of \( y = \cos x \) from \( x = 0 \) to \( x = \pi \):
- At \( x = 0 \), \( y = \cos(0) = 1 \)
- At \( x = \frac{\pi}{2} \), \( y = \cos\left(\frac{\pi}{2}\right) = 0 \)
- At \( x = \pi \), \( y = \cos(\pi) = -1 \)

The area we are interested in is from \( x = 0 \) to \( x = \frac{\pi}{2} \) where \( y \) is positive, and from \( x = \frac{\pi}{2} \) to \( x = \pi \) where \( y \) is negative.

### Step 3: Set up the integral
The area \( A \) can be calculated using the integral:
\[ A = \int_{0}^{\pi} |y| \, dx \]
Since \( y = \cos x \) is positive from \( 0 \) to \( \frac{\pi}{2} \) and negative from \( \frac{\pi}{2} \) to \( \pi \), we can split the integral:
\[ A = \int_{0}^{\frac{\pi}{2}} \cos x \, dx + \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx \]

### Step 4: Calculate the first integral
Calculate \( \int_{0}^{\frac{\pi}{2}} \cos x \, dx \):
\[ \int \cos x \, dx = \sin x \]
Evaluating from \( 0 \) to \( \frac{\pi}{2} \):
\[ \left[ \sin x \right]_{0}^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

### Step 5: Calculate the second integral
Calculate \( \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx \):
\[ \int -\cos x \, dx = -\sin x \]
Evaluating from \( \frac{\pi}{2} \) to \( \pi \):
\[ \left[-\sin x\right]_{\frac{\pi}{2}}^{\pi} = -\sin(\pi) - (-\sin\left(\frac{\pi}{2}\right)) = 0 - (-1) = 1 \]

### Step 6: Add the areas
Now, add the areas from both integrals:
\[ A = 1 + 1 = 2 \]

### Conclusion
The area of the region bounded by the curve \( y = \cos x \), the X-axis, \( x = 0 \), and \( x = \pi \) is \( 2 \) square units.

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