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# Sketch the graph of y=| x+3 | and evaluate$\Large \int\limits_{-6}^0\normalsize| x+3 |dx$

This question has appeared in model paper 2012

Toolbox:
• $\large\frac{d}{dx}$$(x^n)=nx^{n-1} Step 1: y=\mid x+3\mid=\left\{\begin{array}{1 1}-(x+3)\;for\; x<-3\\x+3\;for\; x\geq -3\end{array}\right. When x<-3 y=-x-3 When x=-4, y=4-3=1 When x=-5, y=5-3=2 When x=-6, y=6-3=3 Step 2: When x\geq -3 x=-1\Rightarrow y=2 x=-2\Rightarrow y=1 x=-3\Rightarrow y=0 Step 3: Required Area=Area of region ABC+Area of region OAD \qquad\qquad\quad=\int\limits_{-6}^{-3}\mid x+3\mid dx+\int \limits_{-3}^{0}\mid x+3\mid dx \qquad\qquad\quad=\int\limits_{-6}^{-3}(-x-3)dx+\int \limits_{-3}^0(x+3)dx \qquad\qquad\quad=\big[\large\frac{-x^2}{2}$$-3x\big]_{-6}^{-3}+\big[\large\frac{x^2}{2}+3x\big]_{-3}^0$
Step 4:
On applying limits we get,
$\qquad\qquad\quad=\big[(-\large\frac{9}{2}$$+9)-(-18+18)\big]+\big[\large\frac{9}{2}\big]$
$\qquad\qquad\quad=\large\frac{9}{2}+\frac{9}{2}$
$\qquad\qquad\qquad=9$sq.units