Using integration, find the area of the region bounded by the lines y=1|x+1|, x=-2, x=3 and y=0.

Given the boundaries of the area to be found are,


• The line equation is y = 1 + |x+1|


• The y= 0, x-axis


• x = -2 (a line parallel toy-axis)


• x = 3 (a line parallel toy-axis)



Consider the given line is


y = 1 + |x+1|


this can be written as


y = 1 +(x+1), when x+1≥0 (or) y = 1 –(x+1), when x+1<0


y = x+2, when x≥ -1 (or) y = y = -x, when x<-1 ----(1)


Thus the given boundaries are,


• The line y = 1 +|x+1|.


• x=-2 is parallel toy-axis at -2 units away from the y-axis.


• x=3 is parallel toy-axis at 3 units away from the y-axis.


• y = 0, the x-axis.


The four vertices of the region are,


Point A, where the x-axis and x=3 meet i.e. A(3,0).


Point B, where the line y = 1 +|x+1| and x=3 meet.


Point C, where the line y = 1 +|x+1| and x=-2 meet.


Point D, where the x-axis and x=-2 meet i.e. D(-2,0).


Area of the required region = Area of ABCD.


From (1) we can clearly say that, the area of ABCD has to be divided into twopieces i.e. area under CDFE and EFAB as the line equations changes at x = -1.





[Using the formula and ]




The Area of the required region sq. units.


1