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How to solve this integral concept 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.
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1 Answer

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.