The area of the region bounded by the function `f(x) = x^(3)` , the x-axis and the lines `x = -1` and x = 1 is equal to

Options

A. `(1)/(4)` sq. unit
B. `(1)/(2)` sq.unit
C. `1` sq. units
D. `(1)/(8)` sq. unit

Answer

To find the area of the region bounded by the function \( f(x) = x^3 \), the x-axis, and the lines \( x = -1 \) and \( x = 1 \), we can follow these steps:

### Step 1: Understand the Area to be Calculated
We need to find the area between the curve \( y = x^3 \) and the x-axis from \( x = -1 \) to \( x = 1 \).

### Step 2: Set Up the Integral
Since the function \( f(x) = x^3 \) is symmetric about the y-axis, we can calculate the area from \( x = 0 \) to \( x = 1 \) and then double it to find the total area.

The area \( A \) can be expressed as:
\[ A = 2 \int_{0}^{1} x^3 \, dx \]

### Step 3: Calculate the Integral
Now, we compute the integral:
\[ \int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4} \]
Evaluating this from \( 0 \) to \( 1 \):
\[ \int_{0}^{1} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{1} = \frac{1^4}{4} - \frac{0^4}{4} = \frac{1}{4} \]

### Step 4: Multiply by 2
Now, we multiply the result by 2 to account for the area from \( x = -1 \) to \( x = 0 \):
\[ A = 2 \times \frac{1}{4} = \frac{1}{2} \]

### Final Answer
Thus, the area of the region bounded by the function \( f(x) = x^3 \), the x-axis, and the lines \( x = -1 \) and \( x = 1 \) is:
\[ \boxed{\frac{1}{2}} \]