What is the definite integral of zero?

Mary Jackson

Mary Jackson

Answered question

2021-12-20

What is the definite integral of zero?

Answer & Explanation

Ethan Sanders

Ethan Sanders

Beginner2021-12-21Added 35 answers

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

otoplilp1

Beginner2021-12-22Added 41 answers

Taking the derivative of any constant function is 0, i.e.ddxc=0 So the indefinite integral 0dx 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 αfdx=αfdx? Can`t you say:
0dx=01dx=01dx=0x=0
This gives two conflicting answers. The question is far more complicated that you would first think. But when you say fdx 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:
abαfdx=αabfdx
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 0dx=cc=0
RizerMix

RizerMix

Expert2021-12-29Added 656 answers

Indefinite integrals (anti-derivatoves) are known modulo a constant function. With definite integrals, the case is different:
ab0dt=0
One way to verify that C is the anti-derivative of 0 is simply
ddtC=0

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