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Compute the area bounded by the lines x ...

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

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We have, `x+2y=2 …(i)`
`y-x=1 …(ii)`
and `2x+y=7 …(iii)`
On solving Eqs. (i) and (ii), we get
`y-1(2-2y)=1rArr3y-2=1rArry=1`

`2(y-1)+y=7`
On solving Eqs. (ii) and (iii), we get
`rArr 2y-2+y=7`
`rArr y=3`
On solving Eqs. (i) and (iii) , we get
`2(2-2y)+y=7`
`4-4y+y=7`
`rArr -3y=3`
`rArr y=-1`
`:. " Required area "=int_(-1)^(1)(2-2y)dy+int_(-1)^(3)((7-y))/2dy-int_(1)^(3)(y-1)dy`
`[-2y+(2y^(2))/2]_(-1)^(1)+[(7y)/2-y^(2)/2.2]_(-1)^(3)-[y^(2)/2-y]_(1)^(3)`
`=[-2+2/2-2-2/2]+[21/2-9/4+7/2+1/4]-[9/2-3-1/2+1]`
`=[-4]+[(42-9+14+1)/4]-[(9-6-1+2)/2]`
`=-4+12-2=6" sq units"`
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