The area bounded by the curve `y = (x - 1)^(2) , y = (x + 1)^(2)` and the `x`-axis is

Options

A. (a) `(1)/(3)`
B. (b) `(2)/(3)`
C. (c) `(4)/(3)`
D. (d) `(8)/(3)`

Answer

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

### Step 1: Find the points of intersection of the two curves
We set the two equations equal to each other:
\[ (x - 1)^2 = (x + 1)^2 \]
Expanding both sides:
\[ x^2 - 2x + 1 = x^2 + 2x + 1 \]
Subtracting \( x^2 + 1 \) from both sides:
\[ -2x = 2x \]
Combining like terms gives:
\[ -4x = 0 \implies x = 0 \]
Now, substituting \( x = 0 \) back into either equation to find \( y \):
\[ y = (0 - 1)^2 = 1 \quad \text{or} \quad y = (0 + 1)^2 = 1 \]
Thus, the curves intersect at the point \( (0, 1) \).

### Step 2: Find the points where the curves intersect the x-axis
For \( y = (x - 1)^2 \):
\[ (x - 1)^2 = 0 \implies x - 1 = 0 \implies x = 1 \]
This gives us the point \( (1, 0) \).

For \( y = (x + 1)^2 \):
\[ (x + 1)^2 = 0 \implies x + 1 = 0 \implies x = -1 \]
This gives us the point \( (-1, 0) \).

### Step 3: Set up the integral for the area
The area bounded by the curves and the x-axis can be calculated as:
\[ \text{Area} = \int_{-1}^{0} (x + 1)^2 \, dx + \int_{0}^{1} (x - 1)^2 \, dx \]

### Step 4: Calculate the first integral
Calculating the first integral:
\[ \int (x + 1)^2 \, dx = \frac{(x + 1)^3}{3} \]
Evaluating from \( -1 \) to \( 0 \):
\[ \left[ \frac{(0 + 1)^3}{3} \right] - \left[ \frac{(-1 + 1)^3}{3} \right] = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \]

### Step 5: Calculate the second integral
Calculating the second integral:
\[ \int (x - 1)^2 \, dx = \frac{(x - 1)^3}{3} \]
Evaluating from \( 0 \) to \( 1 \):
\[ \left[ \frac{(1 - 1)^3}{3} \right] - \left[ \frac{(0 - 1)^3}{3} \right] = \frac{0^3}{3} - \frac{(-1)^3}{3} = 0 - \left(-\frac{1}{3}\right) = \frac{1}{3} \]

### Step 6: Add the areas
Adding both areas together:
\[ \text{Total Area} = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \]

### Conclusion
The area bounded by the curves \( y = (x - 1)^2 \), \( y = (x + 1)^2 \), and the x-axis is \( \frac{2}{3} \).