Question

Compute the area bounded by the lines $x+2y=2,y-x=1$ and $2x+y=7$.

Answer 1

We have,
$x+2y=2....\left(i\right)$
$y-x=1....\left(ii\right)$
and $2x+y=7....\left(iii\right)$
On solving Eqs. (i) and (ii), we get
$y-(2-2y)=1$
$\Rightarrow 3y-2=1$
$\Rightarrow y=1$
$\Rightarrow x=0$
On solving Eqs. (ii) and (iii), we get
$2(y-1)+y=7$
$\Rightarrow 2y-2+y=7$
$\Rightarrow y=3\Rightarrow x=2$
On solving Eqs. (i) and (iii), we get
$2(2-2y)+y=7$
$\Rightarrow 4-4y+y=7$
$\Rightarrow -3y=3$
$\Rightarrow y=-1\Rightarrow x=4$

$\therefore$ Required area $={\int }_{-1}^3\frac{(7-y)}{2}dy-\left[{\int }_{-1}^1\right(2-2y)dy+{\int }_1^3(y-1\left)dy\right]$

$=\frac{1}{2}{(7y-\frac{y^2}{2})}_{-1}^3-\left[\right(2y-y^2{)}_{-1}^1+{(\frac{y^2}{2}-y)}_1^3]$

$=\frac{1}{2}(21-\frac{9}{2}+7+\frac{1}{2})-(2-1+2+1)+(\frac{9}{2}-3-\frac{1}{2}+1)$

$=12-4-2=6sq.units$.