Can someone explain this calculus problem for me?

[sportsfreak0925]

I believe I understand what's going on. It seems pretty straight forward. What I'm having trouble with is the intervals. In example 8 it uses 2 sub-intervals, but in exercise 57 it uses 3. Can someone explain why ? If I know how many intervals to use I can solve these problems. For now I'm stuck.

 

[Otis]

In example 8 it uses 2 sub-intervals, but in exercise 57 it uses 3. Can someone explain why?

Did you look at the graphs for each exercise?

No. 8 uses a restricted domain that goes from one x-intercept to another, creating two sub-regions.

The restricted domain in No. 57 extends beyond the x-intercepts, creating three sub-regions.