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Indefinite and Definite Integrals of Zero

Date: 09/12/2004 at 13:56:15
From: Vando
Subject: Integration

Hello Dr. Math, my question is about the integral of zero.  It's a
constant, because the derivative of a constant is zero, right?  But 
how can this be true, because I also was taught that an integral is
the area under a curve...and there is no area under the line x = 0,
right?  Can you help me understand this seeming paradox?

Date: 09/13/2004 at 03:21:01
From: Doctor Luis
Subject: Re: Integration

Hi Vando,

There are two types of integrals at play here.  Definite integrals
are the ones that describe the actual area under a curve.  Indefinite
integrals are the ones that describe the anti-derivative.

There's no paradox, really.  When speaking of indefinite integrals,
the integral of 0 is just 0 plus the usual arbitrary constant, i.e.,

   | 0 dx  = 0 + C = C

There's no contradiction here.  When evaluating the area under a curve 
f(x), we find the antiderivative F(x) and then evaluate from a to b:

    / b
   | f(x) dx = F(b) - F(a)
  / a

So, for f(x) = 0, we find F(x) = C, and so F(b) - F(a) = C - C = 0.
Thus, the total area is zero, as we expected.

Does it make sense now?

Let us know if you have any more questions.

- Doctor Luis, The Math Forum 

Date: 09/13/2004 at 15:56:09
From: Vando
Subject: Thank you (Integration)

Thank you for making the two types of integral definitions clear! :)
Associated Topics:
High School Calculus

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