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Provide solution for RD Sharma maths class 12 chapter Areas of Bounded Region exercise 20.1 question 18

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Answer:

9\; s q \cdot \text { units }

Hint:

Break the limit and find integral

Given:

y=\left | x+3 \right | and evaluate \int_{-6}^{0}|x+3| d x  What represent of graph

Solution:

Let's draw graph, y=\left | x+3 \right |

        y=|x+3|=\left\{\begin{array}{cc} x+3 & \text { for } x+3 \geq 0 \\ -(x+3) & \text { for } x+3<0 \end{array}\right.

        \left\{\begin{array}{c} x+3 \quad \text { for } x \geq-3 \\ -(x+3) \text { for } x+3<-3 \end{array}\right.

                                                    \left[\because \int_{a}^{b} x^{n} d x=\left[\frac{x^{n+1}}{n+1}\right]_{a}^{b}\right]

Required area  

        \begin{aligned} &=\int_{-6}^{0}|x+3| d x \\\\ &=\int_{-6}^{-3}|x+3| d x-=\int_{-3}^{0}|x+3| d x \\\\ &=\int_{-6}^{-3}-(x+3) d x-\int_{-3}^{0}(x+3) d x \end{aligned}

        \begin{aligned} &=\left[\frac{-x^{2}}{2}-3 x\right]_{-6}^{-3}+\left[\frac{x^{2}}{2}+3 x\right]_{-3}^{0} \\\\ &=\left[\frac{-(-3)^{2}}{2}-3(-3)\right]_{-6}^{-3}-\left[\frac{-(-6)^{2}}{2}-3(-6)\right]+\left[\frac{0^{2}}{2}+3(0)\right]-\left[\frac{(-3)^{2}}{2}+3(-3)\right] \end{aligned}

        \begin{aligned} &=\frac{-9}{2}-(-9)-\frac{-36}{2}-(-18)+\left[0+\left(\frac{-9}{2}+9\right)\right] \\\\ &=\frac{-9}{2}+9+0-\frac{9}{2}+9 \\\\ &=-9+18 \\\\ &=9 \text { sq.units } \end{aligned}

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