Question

The area of the region bounded by the curve $y=|x^3-4x^2+3x|$ and the $X$- axis, $0\le x\le 3$, is

• A. $\frac{37}{6}$
• B. $\frac{9}{4}$
• C. $\frac{37}{12}$
• D. $0$

Answer 1

Answer: (C)

We have , $y=\left|x^3-4x^2+3x\right|$
Area of region bounded by $y\le f\left(x\right)$, X- axis
and $0\le x\le 3$ is $\underset{0}{\overset{3}{\int }}\left|x^3-4x^2+3x\right|dx$
$\underset{0}{\overset{1}{\int }}\left(x^3-4x^2+3x\right)-\underset{1}{\overset{3}{\int }}\left(x^3-4x^2+3x\right)dx$
$={\left[\frac{x^4}{4}-\frac{2x^3}{3}+\frac{3x^2}{2}\right]}_0^1-{\left[\frac{x^4}{4}-\frac{4x^3}{3}+\frac{3x^2}{2}\right]}_1^3$
$\left[\left(\frac{1}{4}-\frac{4}{3}+\frac{3}{2}\right)-\left(0\right)\right]$
$-\left[\left(\frac{81}{4}-\frac{108}{3}+\frac{27}{2}\right)-\left(\frac{1}{4}-\frac{4}{3}+\frac{8}{2}\right)\right]$
$=\frac{37}{12}$