Find the area bounded by the parabola `y=x^(2), ` the x -axis and the lines `x=-1, x=2` .

Answer

To find the area bounded by the parabola \( y = x^2 \), the x-axis, and the lines \( x = -1 \) and \( x = 2 \), we can follow these steps:

### Step 1: Identify the area to be calculated
We need to find the area between the curve \( y = x^2 \), the x-axis, and the vertical lines \( x = -1 \) and \( x = 2 \).

### Step 2: Set up the integral for the area
The area \( A \) can be calculated as the sum of two areas: one from \( x = -1 \) to \( x = 0 \) and the other from \( x = 0 \) to \( x = 2 \).

\[ A = A_1 + A_2 \]

Where:
- \( A_1 \) is the area from \( x = -1 \) to \( x = 0 \)
- \( A_2 \) is the area from \( x = 0 \) to \( x = 2 \)

### Step 3: Calculate \( A_1 \)
The area \( A_1 \) can be calculated using the integral:

\[ A_1 = \int_{-1}^{0} x^2 \, dx \]

Calculating the integral:

\[ A_1 = \left[ \frac{x^3}{3} \right]_{-1}^{0} = \left( \frac{0^3}{3} - \frac{(-1)^3}{3} \right) = 0 - \left( -\frac{1}{3} \right) = \frac{1}{3} \]

### Step 4: Calculate \( A_2 \)
Now, we calculate the area \( A_2 \):

\[ A_2 = \int_{0}^{2} x^2 \, dx \]

Calculating the integral:

\[ A_2 = \left[ \frac{x^3}{3} \right]_{0}^{2} = \left( \frac{2^3}{3} - \frac{0^3}{3} \right) = \frac{8}{3} - 0 = \frac{8}{3} \]

### Step 5: Find the total area
Now, we can find the total area \( A \):

\[ A = A_1 + A_2 = \frac{1}{3} + \frac{8}{3} = \frac{9}{3} = 3 \]

### Final Answer
The total area bounded by the parabola \( y = x^2 \), the x-axis, and the lines \( x = -1 \) and \( x = 2 \) is:

\[ \boxed{3} \text{ square units} \]

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