If the area bounded by the parabola `y=2-x^(2)` and the line `y=-x` is `(k)/(2)`sq. units, then the value of 2k is equal to

Options

A. 9
B. 27
C. 18
D. 32

Answer

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

### Step 1: Find the Points of Intersection
To find the points of intersection, we set the equations equal to each other:

\[ 2 - x^2 = -x \]

Rearranging gives us:

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

### Step 2: Solve the Quadratic Equation
Now, we will factor the quadratic equation:

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

This gives us the solutions:

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

### Step 3: Determine the Corresponding y-values
Next, we will find the corresponding \( y \)-values for these \( x \)-values using either of the original equations. Using \( y = -x \):

- For \( x = 2 \):
\[ y = -2 \]
So, the point is \( (2, -2) \).

- For \( x = -1 \):
\[ y = 1 \]
So, the point is \( (-1, 1) \).

### Step 4: Set Up the Integral for the Area
The area \( A \) between the curves from \( x = -1 \) to \( x = 2 \) can be calculated using the integral:

\[ A = \int_{-1}^{2} \left( (2 - x^2) - (-x) \right) \, dx \]

This simplifies to:

\[ A = \int_{-1}^{2} (2 - x^2 + x) \, dx \]

### Step 5: Calculate the Integral
Now, we will compute the integral:

\[ A = \int_{-1}^{2} (2 + x - x^2) \, dx \]

Calculating the integral:

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

### Step 6: Evaluate the Integral at the Bounds
First, evaluate at \( x = 2 \):

\[ = 2(2) + \frac{2^2}{2} - \frac{2^3}{3} = 4 + 2 - \frac{8}{3} = 6 - \frac{8}{3} = \frac{18}{3} - \frac{8}{3} = \frac{10}{3} \]

Now evaluate at \( x = -1 \):

\[ = 2(-1) + \frac{(-1)^2}{2} - \frac{(-1)^3}{3} = -2 + \frac{1}{2} + \frac{1}{3} = -2 + \frac{3}{6} + \frac{2}{6} = -2 + \frac{5}{6} = -\frac{12}{6} + \frac{5}{6} = -\frac{7}{6} \]

### Step 7: Combine the Results
Now, we combine the results:

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

To add these fractions, we find a common denominator (which is 6):

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

### Step 8: Relate to Given Area
According to the problem, the area is given as \( \frac{k}{2} \). Thus:

\[ \frac{9}{2} = \frac{k}{2} \]

This implies:

\[ k = 9 \]

### Step 9: Find the Value of \( 2k \)
Finally, we need to find \( 2k \):

\[ 2k = 2 \times 9 = 18 \]

### Final Answer
Thus, the value of \( 2k \) is:

\[ \boxed{18} \]