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Q.
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
KVPYKVPY 2016
Solution:
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 $\int\limits_{0}^{3} \left|x^{3}-4x^{2}+3x\right|dx $
$\int\limits_{0}^{1} \left(x^{3}-4x^{2}+3x\right)-\int\limits_{1}^{3} \left(x^{3}-4x^{2}+3x\right)dx$
$=\left[\frac{x^{4}}{4}-\frac{2x^{3}}{3}+\frac{3x^{2}}{2}\right]_{0}^{1}-\left[\frac{x^{4}}{4}-\frac{4x^{3}}{3}+\frac{3x^{2}}{2}\right]_{1}^{3}$
$\left[\left(\frac{1}{4}-\frac{4}{3}+\frac{3}{2}\right)-\left(0\right)\right]$
$-\left[\left(\frac{81}{4}-\frac{108}{3}+\frac{27}{2}\right)-\left(\frac{1}{4}-\frac{4}{3}+\frac{8}{2}\right)\right]$
$=\frac{37}{12}$