Q.7. Find the finite area enclosed between the line $y = 0$ and the curve

$$y = x^2 - 4|x| - 12$$

Answer Choices

• $\frac{128}{3}$
• $\frac{176}{3}$
• $\frac{256}{3}$
• 108
• 144
• 288

Solution

We first solve for the roots of the equation by setting $y = 0$:

$$x^2 - 4|x| - 12 = 0$$

The roots are $x = -6$ and $x = 6$. We now set up the integral to find the enclosed area:

$$\text{Area} = \int_{-6}^0 (x^2 + 4x - 12) \, dx + \int_0^6 (x^2 - 4x - 12) \, dx$$

After evaluating the integrals, we find that the total area is:

$$\text{Total Area} = 144$$

Thus, the finite area is 144.