Area bounded by curve `y=sinx`, whit x-axis, when x varies from 0 to `(pi)/(2)` is :-

Options

A. 1 unit
B. 2 units
C. 3 unit
D. 0

Answer

To find the area bounded by the curve \( y = \sin x \), the x-axis, and the vertical lines \( x = 0 \) and \( x = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Set up the integral
The area \( A \) under the curve from \( x = 0 \) to \( x = \frac{\pi}{2} \) can be expressed as an integral:
\[ A = \int_{0}^{\frac{\pi}{2}} \sin x \, dx \]

### Step 2: Integrate the function
We need to find the integral of \( \sin x \). The integral of \( \sin x \) is:
\[ \int \sin x \, dx = -\cos x + C \]
where \( C \) is the constant of integration.

### Step 3: Evaluate the definite integral
Now we will evaluate the definite integral from \( 0 \) to \( \frac{\pi}{2} \):
\[ A = \left[-\cos x\right]_{0}^{\frac{\pi}{2}} = -\cos\left(\frac{\pi}{2}\right) - \left(-\cos(0)\right) \]

### Step 4: Substitute the limits
Now we substitute the limits into the expression:
- \( \cos\left(\frac{\pi}{2}\right) = 0 \)
- \( \cos(0) = 1 \)

Thus, we have:
\[ A = -0 - (-1) = 1 \]

### Conclusion
The area bounded by the curve \( y = \sin x \), the x-axis, and the lines \( x = 0 \) and \( x = \frac{\pi}{2} \) is:
\[ \boxed{1} \text{ unit} \]