What is the area bounded by the function f(x) and the x-axis?

Calculation:

The function is \( y = x^2 - 1 \), and we need to find the area bounded by the curve and the x-axis between x = -1  and x = 1 .

The required area is given by the definite integral of the function from  -1  to 1 :

\( \text{Area} = \int_{-1}^{1} (x^2 - 1) \, dx \)

\( \int (x^2 - 1) \, dx = \frac{x^3}{3} - x \)

Evaluate the integral from  -1 to  1 :

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

\( = \left(\frac{1}{3} - 1\right) - \left(\frac{-1}{3} + 1\right) = \left(\frac{1}{3} - \frac{3}{3}\right) - \left(\frac{-1}{3} + \frac{3}{3}\right) = \frac{2}{3} - \frac{2}{3} = \frac{4}{3} \)

∴ The area is \( \frac{4}{3} \) square units.

Hence, the correct answer is Option 3.