Question

Let $f(x)=x^3+3x+2$ and $g(x)$ be the inverse of it. Then he area bounded by $g(x)$, the $x$-axis, and the ordinate at $x=-2$ and $x=6$ is

• A. $1/4$ sq. units
• B. $4/3$ sq. units
• C. $5/4$ sq. unit
• D. $7/3$ sq. units

Answer 1

Answer: (C)

The required area will be equal to the area enclosed by $y=f(x)$, $y$-axis between the abscissa at $y=-2$ and $y=6$

Hence, $A=\underset{0}{\overset{1}{\int }}\left(6-f\left(x\right)\right)dx+\underset{-1}{\overset{0}{\int }}\left(f\left(x\right)-\left(-2\right)\right)dx$
$=\underset{0}{\overset{1}{\int }}\left(4-x^3-3x\right)dx+\underset{-1}{\overset{0}{\int }}\left(x^3+3x+4\right)dx$
$=\frac{5}{4}$ sq units