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Q.
Area bounded by the lines $y =| x |-2$ and $y =1-| x -1|$ is equal to
Application of Integrals
Solution:
We have, $y=-x-2\,\,\,\,\,\,\,\,\,\,\,...(i)$
$y=x-2\,\,\,\,\,\,\,\,\,\,\,...(ii)$
$y=2-x\,\,\,\,\,\,\,\,\,\,\,...(iii)$
$y=x\,\,\,\,\,\,\,\,\,\,\,...(iv)$
Solving (iii) and (iv), we get $A (1,1)$
Solving (i) and (iv), we get $D(-1,-1)$
Required area $=$ area of $\Delta AOB +$ area of $\Delta OCB +$ area of $\Delta OCD$
Area of $\Delta AOB =\int\limits_{0}^{1} x d x+\int\limits_{1}^{2}(2-x)\, d x$
$=\left[\frac{x^{2}}{2}\right]_{0}^{1}+\left[2 x-\frac{x^{2}}{2}\right]_{1}^{2}=\frac{1}{2}+\left[4-\frac{4}{4}\right]-\left[2-\frac{1}{2}\right]$
$=\frac{1}{2}+\frac{4}{2}-\frac{3}{2}=1$ sq. unit
Area of $\Delta OCB =\left|\int\limits_{0}^{2}( x -2) d x \right|$
$=\left|\left[\frac{ x ^{2}}{2}-2 x \right]_{0}^{2}\right|=2 sq .$ units
Area of $\Delta OCD =\mid \int\limits_{-1}^{0}(- x -2) d x -\int\limits_{-1}^{0} x \,d x$
$=\left|-\left[\frac{ x ^{2}}{2}+2 x \right]_{-1}^{0}-\left[\frac{ x ^{2}}{2}\right]_{-1}^{0}\right|=1$ sq. unit
Required area $=1+2+1=4$ sq. units
