Consider the curves `C_(1) = y - 4x + x^(2) = 0` and `C_(2) = y - x^(2) + x = 0`
The ratio in which the x-axis divide the area of the region bounded by the curves `C_(1) = 0` and `C_(2) = 0` is

Options

A. `(125)/(121)`
B. `(121)/(4)`
C. `(21)/(125)`
D. `(121)/(6)`

Answer

To solve the problem, we need to find the area bounded by the curves \( C_1 \) and \( C_2 \) and then determine the ratio in which the x-axis divides this area.

### Step 1: Identify the curves
The equations of the curves are:
- \( C_1: y = 4x - x^2 \)
- \( C_2: y = x^2 - x \)

### Step 2: Find the points of intersection
To find the points where the curves intersect, we set the equations equal to each other:
\[ 4x - x^2 = x^2 - x \]
Rearranging gives:
\[ 2x^2 - 5x = 0 \]
Factoring out \( x \):
\[ x(2x - 5) = 0 \]
Thus, the solutions are:
\[ x = 0 \quad \text{and} \quad x = \frac{5}{2} \]

### Step 3: Calculate the area between the curves
The area \( A \) between the curves from \( x = 0 \) to \( x = \frac{5}{2} \) can be found using the integral:
\[ A = \int_{0}^{\frac{5}{2}} (C_1 - C_2) \, dx \]
Substituting the equations of the curves:
\[ A = \int_{0}^{\frac{5}{2}} \left( (4x - x^2) - (x^2 - x) \right) \, dx \]
This simplifies to:
\[ A = \int_{0}^{\frac{5}{2}} (5x - 2x^2) \, dx \]

### Step 4: Evaluate the integral
Calculating the integral:
\[ A = \left[ \frac{5}{2}x^2 - \frac{2}{3}x^3 \right]_{0}^{\frac{5}{2}} \]
Calculating at the upper limit:
\[ = \frac{5}{2} \left( \frac{5}{2} \right)^2 - \frac{2}{3} \left( \frac{5}{2} \right)^3 \]
Calculating each term:
\[ = \frac{5}{2} \cdot \frac{25}{4} - \frac{2}{3} \cdot \frac{125}{8} \]
\[ = \frac{125}{8} - \frac{250}{24} \]
Finding a common denominator (24):
\[ = \frac{375}{24} - \frac{250}{24} = \frac{125}{24} \]

### Step 5: Find the areas above and below the x-axis
1. **Area \( A_1 \)** (above the x-axis):
- From \( x = 0 \) to \( x = 1 \):
\[ A_1 = \int_{0}^{1} (4x - x^2) \, dx = \left[ 2x^2 - \frac{1}{3}x^3 \right]_{0}^{1} = 2 - \frac{1}{3} = \frac{5}{3} \]

2. **Area \( A_2 \)** (below the x-axis):
- From \( x = 1 \) to \( x = \frac{5}{2} \):
\[ A_2 = \int_{1}^{\frac{5}{2}} (x^2 - x - (4x - x^2)) \, dx = \int_{1}^{\frac{5}{2}} (-3x + 2x^2) \, dx \]
Evaluating this integral:
\[ = \left[ -\frac{3}{2}x^2 + \frac{2}{3}x^3 \right]_{1}^{\frac{5}{2}} \]
Calculating at the limits gives:
\[ = \left( -\frac{3}{2} \cdot \frac{25}{4} + \frac{2}{3} \cdot \frac{125}{8} \right) - \left( -\frac{3}{2} + \frac{2}{3} \right) \]

### Step 6: Calculate the ratio of areas
Finally, we find the ratio:
\[ \text{Ratio} = \frac{A_1}{A_2} \]