A circel 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 :

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 the circle centered at the origin with a radius of \( \pi \) 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 and intersects the circle at certain points.

### Step 2: Find the Points of Intersection
To find the points where the curve intersects the circle, we set \( y = \sin x \) into the circle's equation:
\[ x^2 + (\sin x)^2 = \pi^2 \]
This equation is complex to solve analytically, but we can note that the intersections occur in the range where \( \sin x \) is defined, particularly between \( -\pi \) and \( \pi \).

### Step 3: Calculate the Area of the Semicircle
The area of the entire circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
For our circle with radius \( \pi \):
\[ \text{Area} = \pi (\pi^2) = \pi^3 \]
Since we only need the area of the upper half (semicircle), we divide this area by 2:
\[ \text{Area of semicircle} = \frac{\pi^3}{2} \]

### Step 4: Area Above the Curve
The curve \( y = \sin x \) divides the semicircle into two equal areas since it is symmetric about the y-axis. Therefore, the area of the upper part of the semicircle above the curve \( y = \sin x \) is equal to half of the semicircle's area:
\[ \text{Area of upper part} = \frac{\pi^3}{2} \]

### Final Answer
Thus, the area of the upper part of the circle divided by the curve \( y = \sin x \) is:
\[ \frac{\pi^3}{2} \]

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