Why do we say that when a definite integral equals infinity it diverges?

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When we are in the process of calculating an improper integral,specifically the type II ones,we say that if the limit does not exist then the integral diverges.

But isn't it rational to say that an infinite sum (the definite integral) is equal to infinity?Why can't we say that this sum=infinity?Is it because the idea of an infinite area is geometrically quite abstract?

Thank you!