The area bounded by the curve `x=2-y-y^(2)` and Y-axis is

Options

A. `(3)/(2)` sq units
B. `(5)/(2)` sq units
C. `(9)/(2)` sq units
D. None of these

Explanation

To find the area bounded by the curve \( x = 2 - y - y^2 \) and the Y-axis, we will follow these steps:

### Step 1: Find the intersection points of the curve and the Y-axis
The Y-axis corresponds to \( x = 0 \). We set the equation of the curve equal to zero:

\[ 0 = 2 - y - y^2 \]

Rearranging gives:

\[ y^2 + y - 2 = 0 \]

### Step 2: Factor the quadratic equation
We can factor the quadratic equation:

\[ (y + 2)(y - 1) = 0 \]

This gives us the solutions:

\[ y = -2 \quad \text{and} \quad y = 1 \]

### Step 3: Set up the integral for the area
The area \( A \) bounded by the curve and the Y-axis can be expressed as:

\[ A = \int_{y_1}^{y_2} x \, dy \]

where \( y_1 = -2 \) and \( y_2 = 1 \). The expression for \( x \) from the curve is \( x = 2 - y - y^2 \). Therefore, we have:

\[ A = \int_{-2}^{1} (2 - y - y^2) \, dy \]

### Step 4: Integrate the function
Now we will compute the integral:

\[ A = \int_{-2}^{1} (2 - y - y^2) \, dy \]

Calculating the integral:

\[ = \left[ 2y - \frac{y^2}{2} - \frac{y^3}{3} \right]_{-2}^{1} \]

### Step 5: Evaluate the definite integral
Now we will evaluate the integral at the limits:

1. Evaluate at \( y = 1 \):

\[ = 2(1) - \frac{(1)^2}{2} - \frac{(1)^3}{3} = 2 - \frac{1}{2} - \frac{1}{3} \]

Finding a common denominator (6):

\[ = \frac{12}{6} - \frac{3}{6} - \frac{2}{6} = \frac{12 - 3 - 2}{6} = \frac{7}{6} \]

2. Evaluate at \( y = -2 \):

\[ = 2(-2) - \frac{(-2)^2}{2} - \frac{(-2)^3}{3} = -4 - 2 + \frac{8}{3} \]

Finding a common denominator (3):

\[ = -\frac{12}{3} - \frac{6}{3} + \frac{8}{3} = -\frac{12 + 6 - 8}{3} = -\frac{10}{3} \]

### Step 6: Combine the results
Now, we combine the results from both evaluations:

\[ A = \left( \frac{7}{6} \right) - \left( -\frac{10}{3} \right) \]

Converting \(-\frac{10}{3}\) to sixths:

\[ = \frac{7}{6} + \frac{20}{6} = \frac{27}{6} = \frac{9}{2} \]

### Final Answer
Thus, the area bounded by the curve \( x = 2 - y - y^2 \) and the Y-axis is:

\[ \boxed{\frac{9}{2}} \]