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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
http://mathforum.org/dr.math/

```

```
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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