The area bounded by `y=sin^(-1)x`, y-axis and `|y|=(pi)/(2)` is

Options

A. 2
B. `pi`
C. `2pi`
D. 1

Answer

To find the area bounded by the curves \( y = \sin^{-1}(x) \), the y-axis, and \( |y| = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Identify the curves and their intersections
The curves we are dealing with are:
1. \( y = \sin^{-1}(x) \)
2. The y-axis, which is \( x = 0 \)
3. The horizontal lines \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \)

The function \( y = \sin^{-1}(x) \) is defined for \( x \) in the interval \([-1, 1]\) and takes values in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

### Step 2: Determine the area of interest
We are interested in the area bounded by \( y = \sin^{-1}(x) \) from \( x = 0 \) to \( x = 1 \) and the lines \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \). Since the area is symmetric about the x-axis, we can calculate the area in the first quadrant and then double it.

### Step 3: Set up the integral
To find the area, we will integrate with respect to \( y \) from \( 0 \) to \( \frac{\pi}{2} \). The relationship between \( x \) and \( y \) is given by \( x = \sin(y) \). Therefore, the area \( A \) can be expressed as:
\[ A = 2 \int_{0}^{\frac{\pi}{2}} x \, dy = 2 \int_{0}^{\frac{\pi}{2}} \sin(y) \, dy \]

### Step 4: Compute the integral
Now we compute the integral:
\[ \int \sin(y) \, dy = -\cos(y) \]
Evaluating this from \( 0 \) to \( \frac{\pi}{2} \):
\[ \int_{0}^{\frac{\pi}{2}} \sin(y) \, dy = \left[-\cos(y)\right]_{0}^{\frac{\pi}{2}} = -\cos\left(\frac{\pi}{2}\right) + \cos(0) = -0 + 1 = 1 \]

### Step 5: Calculate the total area
Now substituting back into our area formula:
\[ A = 2 \cdot 1 = 2 \]

### Final Answer
The area bounded by the curves is \( 2 \) square units.