A circle centered at origin and having radius `pi` units is divided by the curve `y= sin x` in two parts. Then area of the upper part equals to : (a)`(pi^(2))/(2)` (b)`(pi^(3))/(4)` (c)`(pi^(3))/(2)` (d)`(pi^(3))/(8)`

Options

A. `(pi^(2))/(2)`
B. `(pi^(3))/(4)`
C. `(pi^(3))/(2)`
D. `(pi^(3))/(8)`

Answer

To find the area of the upper part of a circle centered at the origin with radius \(\pi\) units, divided by the curve \(y = \sin x\), we can follow these steps:

### Step 1: Understand the Circle and the Curve
The equation of the circle centered at the origin with radius \(\pi\) is given by:
\[ x^2 + y^2 = \pi^2 \]
The curve \(y = \sin x\) oscillates between -1 and 1. We need to find the area of the upper part of the circle that is above the curve \(y = \sin x\).

### Step 2: Find the Area of the Circle
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting \(r = \pi\):
\[ A = \pi (\pi)^2 = \pi^3 \]

### Step 3: Determine the Area Above the Curve
Since the curve \(y = \sin x\) divides the circle into two symmetrical parts, the area above the curve will be half of the total area of the circle. Therefore, the area of the upper part is:
\[ \text{Area of upper part} = \frac{1}{2} \times \text{Area of circle} = \frac{1}{2} \times \pi^3 = \frac{\pi^3}{2} \]

### Step 4: Conclusion
Thus, the area of the upper part of the circle, which is above the curve \(y = \sin x\), is:
\[ \frac{\pi^3}{2} \]

### Final Answer
The correct option is (c) \(\frac{\pi^3}{2}\).

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