Question

Sketch the graph y =|x + 1|. Evaluate ∫|x + 1| dx, x ∈ [1, -3] What does this value represent on the graph ?

Answer 1

We have, y = |x + 1| = \(\begin{cases} x+1, if x+1\geq0\,i.e.,x\geq-1\\ x+1, if \,x+1<0\,i.e.,x<-1 \end{cases}\)

So, we have  y = x +1 for x ≥ -1 and y = -x - 1 for x < -1. Clearly, y = x + 1 is a straight line cutting x and y-axes at (-1, 0) and (0, 1) respectively. So, y = x + 1, x ≥ -1 represents that portion of the line which lies on the right side of x = -1. Similarly, y = -x - 1, x < -1 represents that part of the line y = -x -1 which is on the left side x = -1. A rough sketch of y = |x + 1| is shown in fig.

Now, 

This value represents the area of the shaded portion shown in figure.