The area bounded by by `y=cosx,y=0,|x|=1` is

Options

A. A) `4sin1`
B. B) `3 sin 1`
C. C) `2 sin 1`
D. D) `sin1`

Answer

To find the area bounded by the curves \( y = \cos x \), \( y = 0 \), and \( |x| = 1 \), we can follow these steps:

### Step 1: Understand the boundaries
The curves given are:
- \( y = \cos x \): This is a cosine wave.
- \( y = 0 \): This is the x-axis.
- \( |x| = 1 \): This means \( x = -1 \) and \( x = 1 \).

### Step 2: Sketch the graph
Draw the graph of \( y = \cos x \) from \( x = -1 \) to \( x = 1 \). The cosine function reaches its maximum at \( x = 0 \) (where \( y = 1 \)) and crosses the x-axis at \( x = \pm \frac{\pi}{2} \).

### Step 3: Identify the area to be calculated
The area we need to calculate is the region between the curve \( y = \cos x \) and the x-axis from \( x = -1 \) to \( x = 1 \).

### Step 4: Set up the integral
Since the area is symmetric about the y-axis, we can calculate the area from \( x = 0 \) to \( x = 1 \) and then double it. The area \( A \) can be expressed as:
\[ A = 2 \int_{0}^{1} \cos x \, dx \]

### Step 5: Calculate the integral
Now, we compute the integral:
\[ \int \cos x \, dx = \sin x \]
Thus, we evaluate:
\[ \int_{0}^{1} \cos x \, dx = \sin x \bigg|_{0}^{1} = \sin(1) - \sin(0) = \sin(1) - 0 = \sin(1) \]

### Step 6: Calculate the total area
Now, substituting back into our area formula:
\[ A = 2 \cdot \sin(1) \]

### Final Answer
The area bounded by the curves is:
\[ A = 2 \sin(1) \]