Using integration, find the area of the region bounded by the following curves after making a rough sketch: `y=1+|x+1|, x=-2, x=3, y=0.`

Explanation

Given equations are:
` y = 1 + |x + 1| `
`y = 1 + x + 1`, if `x + 1≥ 0 `
`y_2 = 2 + x …… (1)`, if `x ≥ - 1`
And `y = 1 – (x + 1)`, if `x + 1 < 0`
` y = 1 – x – 2`, if `x < - 1`
` y_1 = - x …… (2)`, if` x < - 1`
` x = - 2 ...... (3)`
` x = 3 ...... (4)`
`y = 0 ...... (5)`
We have to find the area of shaded region.
Required area = (shaded region `ABCDEA`) = shaded region `ABCFA` + Shaded region `FCDEFC`
`int_-2^-1(-x)dx+int_-1^3(x+2)dx`
`=-[x^2/2]_-2^-1+[x^2/2]_-1^3+2[x]_-1^3`
`=-[(1-4)/2]+[(9-1)/2]+8`
`=3/2+8/2+8`
`=27/2`
hence the area of the shaded region is `27/2` sq.units.