Definition of Definite Integral

Philosophy Forums Science, Technology, and Math
1

I have a question. My Calc textbook is saying that the definition of a definite intergral is:

A function f defined on an interval [a,b] is integrable on [a,b] if there is one and only one number I that satisfies the inequality

L(P) =or< I < U(P) for all partitions P of [a,b],

the number I is called the definite integral.

My question is, how can there be only one number that satisfies this inequality?? If I < U(P), then L(P) =or< I < (I+U(P))/2 < U(P), thereby making it impossible to have only one number satisfying that inequality. I think the text is wrong but school is out right now and I have no one to ask the proper definition of.

What sounds better is L(P) =or< I =or< U(P). Does anyone out there know the proper definition of a definite integral?

2

You’re right, the text is incorrect, and your reasoning is one way to show the definition is ill-conceived. Another way would be to consider the integral of f(x) = 0. All upper and lower partitions are equal to 0, so the integral cannot exist according to the textbook’s definition since there is no number I such that 0 <= I < 0. It has to be 0 <= I <= 0.

3

Thanks Aporia…I appreciate it bud! I’ll change my text.