What is the definite integral of zero?

Question

Ethan Sanders · Accepted Answer

If you mean ∫ab0dx , it is equal to zero.  This can be seen in a number of ways.  Intuitively, the area under the graph of the null function is always zero, no matter over what interval we chose to evaluate it. Therefore ∫ab0dx should be equal to 0, although this isnt

otoplilp1 · Accepted Answer

Taking the derivative of any constant function is 0, i.e.

\frac{d}{d x} c = 0

So the indefinite integral

\int 0 d x

produces the class of constant functions, that is

f (x) = c

for some c
There's something that you have to look at here though, that is "what about the fact

α \int f d x = \int α f d x

? Can`t you say:

\int 0 d x = \int 0 \cdot 1 d x = 0 \int 1 d x = 0 x = 0

This gives two conflicting answers. The question is far more complicated that you would first think. But when you say

\int f d x

and the interval over which you're integrating isn't obvious or defined, what you really mean is "the class of functions that when derived with respect to x produce f The rule stated only applies for definite integrals. That is:

\int_{a}^{b} α f d x = α \int_{a}^{b} f d x

And if you look at textbooks on real analysis (I just looked at Rudin) that's the form in which you will find the theorem.
It should also be noted that the definite integral of 0 over any interval is 0, as

\int 0 d x = c - c = 0

RizerMix · Accepted Answer

Indefinite integrals (anti-derivatoves) are known modulo a constant function. With definite integrals, the case is different:
$\int_{a}^{b} 0 d t = 0$
One way to verify that C is the anti-derivative of 0 is simply
$\frac{d}{d t} C = 0$

What is the definite integral of zero?

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New Questions in Calculus 1