If the area bounded by `y=x, y=sinx and x=(pi)/(2)` is `((pi^(2))/(k)-1)` sq. units then the value of k is equal to

Options

A. 2
B. 3
C. 6
D. 8

Answer

To find the value of \( k \) such that the area bounded by the curves \( y = x \), \( y = \sin x \), and the line \( x = \frac{\pi}{2} \) is given by the expression \( \frac{\pi^2}{k} - 1 \), we can follow these steps:

### Step 1: Set up the area integral
The area \( A \) between the curves \( y = x \) and \( y = \sin x \) from \( x = 0 \) to \( x = \frac{\pi}{2} \) can be expressed as:
\[ A = \int_0^{\frac{\pi}{2}} (x - \sin x) \, dx \]