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Q.
The area bounded by the line $y = 2x -2$ , $y = - x$ and $x$-axis is given by
Application of Integrals
Solution:
We have, $y = 2x - 2\quad\ldots\left(i\right)$
$y=-x\quad\left(ii\right)$
Solving $(i)$ and $(ii)$, we get $x=\frac{2}{3}$, $y=\frac{-2}{3}$
Required area = area of shaded region
$A=\left|\int\limits_{0}^{2/ 3}\left(-x\right)dx\right|+\left|\int\limits_{2 /3}^{1}\left(2x-2\right)dx\right|$
$=\left|\left[\frac{-x^{2}}{2}\right]_{0}^{2 /3}\right|+\left|\left[x^{2}-2x\right]_{2 /3}^{1}\right|=\frac{4}{18}+\frac{1}{9}=\frac{1}{3}$ sq. units
