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,
$f\left(x\right)=x^4-2x^2...\left(i\right)$
$g\left(x\right)=2x^2...\left(ii\right)$
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
$=2\underset{0}{\overset{2}{\int }}\left(g\left(x\right)-f\left(x\right)\right)dx$
$=2\underset{0}{\overset{2}{\int }}\left(2x^2-x^4+2x^2\right)dx$
$=2\underset{0}{\overset{2}{\int }}\left(4x^2-x^4\right)dx$
$=2{\left[\frac{4x^3}{3}-\frac{x^5}{5}\right]}_0^2=2\left(\frac{32}{3}-\frac{32}{5}\right)$
$=\frac{128}{15}$