Question

Draw a rough sketch of the given curve y = 1 + |x +1|, x = –3, x = 3, y = 0 and find the area of the region bounded by them, using integration.

Answer 1

Given equations are y = 1 + |x + 1|

x = –3 and x = 3

y = 0

Taking y = 1 + |x + 1|

⇒ y = 1 + x + 1

⇒ y = x + 2

And y = 1 – x – 1

⇒ y = –x

On solving we get x = –1

Area of the required regions = `int_(-3)^(-1) -x  "d"x + int_(-1)^3 (x + 2)  "d"x`

= `-[x^2/2]_-3^-1 + [x^2/2 + 2x]_1^3`

= `-[1/2 - 9/2] + [(9/2 + 6) - (1/2 - 2)]`

= `-(-4) + [21/2 + 3/2]`

= 4 + 12

= 16 sq.units

Hence, the required area = 16 sq.units