Sketch the graph y = |x + 3|. Evaluate . What does this integral represent on the graph?

Given equations are:

y = |x + 3|

y_{1} = x + 3, if x + 3 ≥ 0

y_{1} = x + 3 …… (1), if x ≥ - 3

And y_{2} = - (x + 3), if x + 3 < 0

y_{2} = - (x + 3) …… (2), if x < - 3

So, equation (1) is straight line that passes thorough ( - 3,0) and (0,3). Equation (2) is a line passing through ( - 3,0). So, the graph of which is as follows:

(As x is between ( - 6, - 3) in first shaded region equation becomes as y_{2} and when x is between ( - 3,0) for the second shaded region equation becomes y_{1})

(from equation (2))

Now integrating by applying power rule, we get

Now applying the limits we get

Hence the value of represents the area of the shaded region (as shown in the graph) and is equal to 9 square units.

18