How to solve this integral concept question

Question

Explain why the total area of a region is 25/3 units, but the value of the integral for the same interval is 16/3 units.

Answer 1

Simple: over part of the interval, if the function values are negative, then the contribution to the definite integral would be negative.  The definite integral only equals the area under a curve if the integrand's values are never negative over that interval.