The area bounded by the curve `|x|=cos^-1y` and the line `|x|=1` and the x-axis is

Options

A. `2 sqrt(1 - cos^(2)1)`
B. sin 1
C. 2 cos 1
D. 2 sin 1

Answer

To find the area bounded by the curve \( |x| = \cos^{-1}(y) \), the line \( |x| = 1 \), and the x-axis, we can follow these steps:

### Step 1: Understand the given equations
The equation \( |x| = \cos^{-1}(y) \) implies that:
- For \( x \geq 0 \), \( x = \cos^{-1}(y) \)
- For \( x < 0 \), \( -x = \cos^{-1}(y) \) or \( x = -\cos^{-1}(y) \)

### Step 2: Rewrite the equation
From the equation \( |x| = \cos^{-1}(y) \), we can express \( y \) in terms of \( x \):
- \( y = \cos(|x|) \)

### Step 3: Determine the boundaries
The lines \( |x| = 1 \) correspond to \( x = 1 \) and \( x = -1 \). The area we are interested in is between these two lines and above the x-axis.

### Step 4: Set up the integral for the area
The area \( A \) can be calculated using the integral:
\[ A = \int_{-1}^{1} \cos(|x|) \, dx \]
Since \( \cos(|x|) = \cos(x) \) for \( x \geq 0 \) and \( \cos(-x) = \cos(x) \) for \( x < 0 \), we can simplify the integral:
\[ A = \int_{-1}^{1} \cos(x) \, dx \]

### Step 5: Evaluate the integral
Now we can evaluate the integral:
\[ A = \int_{-1}^{1} \cos(x) \, dx = \left[ \sin(x) \right]_{-1}^{1} \]
Calculating the limits:
\[ A = \sin(1) - \sin(-1) \]
Since \( \sin(-x) = -\sin(x) \), we have:
\[ A = \sin(1) + \sin(1) = 2\sin(1) \]

### Final Answer
Thus, the area bounded by the curve, the line, and the x-axis is:
\[ A = 2\sin(1) \]

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