question_answer

Using the method of integration, find the area bounded by the curve \[|x|+|y|\,\,=1.\]

Answer:

Given equation of curve is \[|x|+|y|=1\]             \[\Rightarrow \]   \[\pm \,x\pm y=1\]         \[\left[ \because \,\,|x|=\left\{ \begin{matrix}    x,\,\,\text{if}\,\,x\ge 0  \\    -x,\,\,\text{if}\,\,x<0  \\ \end{matrix} \right. \right]\] The above equation represents these four lines,             \[x+y=1\]                                 ? (i)             \[x-y=1\]                                   ? (ii)             \[-x+y=1\]                                ? (iii) and       \[-x-y=1\]                                  ? (iv) The graphical representation of these lines is given below Since, the required area is symmetrical in all the four quadrants.                                     Required area = 4 (Area of region OABO)             \[=4\int_{0}^{1}{ydx=4\int_{0}^{1}{(1-x)}}dx\]           \[[\because \,\,x+y=1]\] \[=4\left[ x-\frac{{{x}^{2}}}{2} \right]_{0}^{1}=4\left[ \left( 1-\frac{{{1}^{2}}}{2} \right)-\left( 0-\frac{{{0}^{2}}}{2} \right) \right]\]  \[=4\left( 1-\frac{1}{2} \right)=4\left( \frac{1}{2} \right)=2\,\text{sq}.\] units