Question

The area bounded by the graphs of functions $f (x) = x^4 - 2x^2$ and $g (x) = 2x^2 $ is

• A. $\frac{121}{15}$
• B. $\frac{124}{15}$
• C. $\frac{128}{15}$
• D. $\frac{131}{15}$

Answer 1

Answer: (C)

We have,
$f\left(x\right) = x^{4} -2x^{2} \quad...\left(i\right) $
$g\left(x\right) = 2x^{2} \quad...\left(ii\right) $
The intersection point of $f \left(x \right)$ and $g \left(x \right)$ are $\left(0, 0\right)\left(2,8\right) $ and $\left( -2, 8\right) $.

Area of shaded region
$ = 2\int\limits_{0}^{2} \left(g\left(x\right) -f\left(x\right)\right)dx $
$ =2\int\limits_{0}^{2} \left(2x^{2} -x^{4} +2x^{2}\right)dx $
$ = 2\int\limits_{0}^{2} \left(4x^{2} -x^{4}\right)dx $
$= 2\left[\frac{4x^{3}}{3} - \frac{x^{5}}{5}\right]_{0}^{2} = 2 \left(\frac{32}{3} - \frac{32}{5}\right) $
$= \frac{128}{15} $