Draw a rough sketch and find the area bounded by the curve `x^(2)=yandx+y=2`.

Answer

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

### Step 1: Identify the curves
The first equation \( x^2 = y \) represents a parabola that opens upwards. The second equation \( x + y = 2 \) can be rearranged to \( y = 2 - x \), which is a straight line with a slope of -1.

### Step 2: Find the points of intersection
To find the area between these two curves, we first need to determine where they intersect. We can do this by substituting \( y \) from the line equation into the parabola equation:

\[ x^2 = 2 - x \]

Rearranging gives us:

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

### Step 3: Solve the quadratic equation
Now, we will factor or use the quadratic formula to solve for \( x \):

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

This gives us the solutions:

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

### Step 4: Find corresponding \( y \) values
Next, we will find the \( y \) values for each \( x \):

1. For \( x = -2 \):
\[ y = 2 - (-2) = 4 \]

2. For \( x = 1 \):
\[ y = 2 - 1 = 1 \]

Thus, the points of intersection are \( (-2, 4) \) and \( (1, 1) \).

### Step 5: Set up the integral for the area
The area \( A \) between the curves from \( x = -2 \) to \( x = 1 \) can be calculated as:

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

### Step 6: Evaluate the integral
Now we will evaluate the integral:

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

Calculating the integral:

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

Now, we will evaluate this expression at the limits:

1. At \( x = 1 \):
\[ 2(1) - \frac{(1)^2}{2} - \frac{(1)^3}{3} = 2 - \frac{1}{2} - \frac{1}{3} = 2 - 0.5 - 0.3333 = 1.1667 \]

2. At \( x = -2 \):
\[ 2(-2) - \frac{(-2)^2}{2} - \frac{(-2)^3}{3} = -4 - 2 + \frac{8}{3} = -6 + 2.6667 = -3.3333 \]

Now, substituting back into the area formula:

\[ A = \left( 1.1667 - (-3.3333) \right) = 1.1667 + 3.3333 = 4.5 \]

### Final Result
Thus, the area bounded by the curve \( x^2 = y \) and the line \( x + y = 2 \) is:

\[ \text{Area} = \frac{9}{2} \text{ square units} \]