The ratio in which the curve `y = x^(2)` divides the region bounded by the curver: `y = sin'((pix)/(2))` and the x-axis as x varies from 0 to 1, is :

Options

A. `2 : pi`
B. `1 : 3`
C. `3 : pi`
D. `(6-pi) : pi`

Answer

To solve the problem, we need to find the ratio in which the curve \( y = x^2 \) divides the area bounded by the curve \( y = \sin\left(\frac{\pi x}{2}\right) \) and the x-axis from \( x = 0 \) to \( x = 1 \).

### Step 1: Find the area under the curve \( y = \sin\left(\frac{\pi x}{2}\right) \) from \( x = 0 \) to \( x = 1 \).

The area \( A_1 \) can be calculated using the definite integral:

\[ A_1 = \int_0^1 \sin\left(\frac{\pi x}{2}\right) \, dx \]

### Step 2: Compute the integral for \( A_1 \).

Using the substitution \( u = \frac{\pi x}{2} \), we have \( du = \frac{\pi}{2} dx \) or \( dx = \frac{2}{\pi} du \). The limits change as follows:
- When \( x = 0 \), \( u = 0 \)
- When \( x = 1 \), \( u = \frac{\pi}{2} \)

Thus, the integral becomes:

\[ A_1 = \int_0^{\frac{\pi}{2}} \sin(u) \cdot \frac{2}{\pi} \, du = \frac{2}{\pi} \left[-\cos(u)\right]_0^{\frac{\pi}{2}} = \frac{2}{\pi} \left[-\cos\left(\frac{\pi}{2}\right) + \cos(0)\right] \]

Calculating this gives:

\[ A_1 = \frac{2}{\pi} \left[0 + 1\right] = \frac{2}{\pi} \]

### Step 3: Find the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 1 \).

The area \( A_2 \) can be calculated using the definite integral:

\[ A_2 = \int_0^1 x^2 \, dx \]

### Step 4: Compute the integral for \( A_2 \).

Calculating this integral gives:

\[ A_2 = \left[\frac{x^3}{3}\right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \]

### Step 5: Find the ratio \( \frac{A_1}{A_2} \).

Now we can find the ratio of the two areas:

\[ \frac{A_1}{A_2} = \frac{\frac{2}{\pi}}{\frac{1}{3}} = \frac{2}{\pi} \cdot 3 = \frac{6}{\pi} \]

### Step 6: Express the ratio in the required format.

The ratio \( A_1 : A_2 \) is:

\[ A_1 : A_2 = 6 - \pi : \pi \]

### Final Answer:

Thus, the ratio in which the curve \( y = x^2 \) divides the region bounded by the curve \( y = \sin\left(\frac{\pi x}{2}\right) \) and the x-axis is:

\[ \boxed{6 - \pi : \pi} \]