The area bounded by the curve `y=|cos^-1(sinx)|+|pi/2-cos^-1(cosx)| and` the x - axis , where `pi/2lexlepi`, is equal to

Options

A. `pi^2`
B. `(pi^2)/2`
C. `(pi^2)/8`
D. `(pi^2)/4`

Answer

To find the area bounded by the curve \( y = |\cos^{-1}(\sin x)| + |\frac{\pi}{2} - \cos^{-1}(\cos x)| \) and the x-axis for \( \frac{\pi}{2} \leq x \leq \pi \), we will follow these steps:

### Step 1: Simplify the expression for \( y \)

We start with the expression:
\[ y = |\cos^{-1}(\sin x)| + |\frac{\pi}{2} - \cos^{-1}(\cos x)| \]

For \( \frac{\pi}{2} \leq x \leq \pi \):
- \( \sin x \) is non-negative, so \( \cos^{-1}(\sin x) \) is valid and \( |\cos^{-1}(\sin x)| = \cos^{-1}(\sin x) \).
- \( \cos x \) is non-positive, thus \( \cos^{-1}(\cos x) = \pi - x \) (since \( \cos^{-1} \) gives the angle in the range \( [0, \pi] \)).

Now, we can rewrite the expression:
\[ y = \cos^{-1}(\sin x) + \left| \frac{\pi}{2} - (\pi - x) \right| \]
This simplifies to:
\[ y = \cos^{-1}(\sin x) + \left| x - \frac{\pi}{2} \right| \]

### Step 2: Evaluate \( y \)

In the interval \( \frac{\pi}{2} \leq x \leq \pi \):
- \( \cos^{-1}(\sin x) = \frac{\pi}{2} - x \) (since \( \sin x \) is positive in this range).
- Thus, we have:
\[ y = \left( \frac{\pi}{2} - x \right) + \left( x - \frac{\pi}{2} \right) = 0 \]

### Step 3: Determine the area under the curve

Since \( y \) simplifies to \( 0 \) in the given interval, the area bounded by the curve and the x-axis is determined by the line segments formed by the endpoints of the interval.

### Step 4: Area Calculation

To find the area, we need to consider the triangle formed by the line \( y = 2x - \pi \) from \( x = \frac{\pi}{2} \) to \( x = \pi \):
- The base of the triangle is from \( x = \frac{\pi}{2} \) to \( x = \pi \), which has a length of \( \pi - \frac{\pi}{2} = \frac{\pi}{2} \).
- The height of the triangle is the value of \( y \) at \( x = \frac{\pi}{2} \):
\[ y\left(\frac{\pi}{2}\right) = 2\left(\frac{\pi}{2}\right) - \pi = 0 \]
and at \( x = \pi \):
\[ y(\pi) = 2\pi - \pi = \pi \]

The area \( A \) of the triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \frac{\pi}{2} \times \pi = \frac{\pi^2}{4} \]

### Final Answer

Thus, the area bounded by the curve and the x-axis is:
\[ \boxed{\frac{\pi^2}{4}} \]