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Integrals versus Antiderivatives


Date: 02/24/2001 at 12:46:11
From: Matt Orseske
Subject: Calculus

What is the difference between an integral and an antiderivative?


Date: 02/25/2001 at 15:01:59
From: Doctor Fenton
Subject: Re: Calculus

Hi Matt,

Thanks for writing to Dr. Math. This is a very good question, since 
the difference seems to be glossed over in many courses today.

An integral is a number associated with a function, what is usually 
called a "definite integral." It is defined by a limiting process. You 
generally learn Riemann integration first, but if you continue in 
mathematics, you may encounter other types of integrals such as the 
Lebesgue integral, the Denjoy integral, the Daniell integral, etc.

But each shares this property of associating with each suitable 
function on a given domain a number. The definition of this process 
has nothing to do with differentiation. For Riemann integrals in one 
dimension, you form "Riemann sums" by dividing the domain into small 
intervals and approximating the area bounded by the x-axis, the 
vertical lines at the ends of the subinterval, and the function graph 
by the area of a rectangle, bounded by the x-axis, the vertical lines, 
and a horizontal line at the height of f(x) at some point in the 
subinterval.

For Riemann integrals, there is the remarkable Fundamental Theorem of 
Calculus, which says that if F(x) is an anti-derivative of f(x), then

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

This says that we can calculate the number (the "integral") associated 
with f(x) on the interval [a,b] if we know an antiderivative F(x). For 
this reason, many people indicate the antidifferentiation process by a 
similar notation, and call it "indefinite integration". They write:

     /
     | f(x)dx = F(x)     (or F(x)+C , since any two antiderivatives 
     /                     differ by a constant)

In higher dimensions, there is no Fundamental Theorem of Calculus 
connecting multiple integrals with partial derivatives, so there isn't 
an "antidifferentiation" process for functions of several variables. 
The closest correspondence would probably the Divergence Theorem or 
Stokes Theorem, which connect integrals of certain specific forms over 
a volume or surface to lower dimensional integrals of related 
integrands over the boundaries of those volumes or surfaces.

Other integrals such as the Lebesgue integral do not have anything 
similar to the Fundamental Theorem of Calculus. For computation, 
however, since a Riemann-integrable function on a bounded interval is 
also Lebesgue-integrable, and their values are equal, Lebesgue 
integrals are often calculated as Riemann integrals by using the FTC. 
Multiple integrals can often be evaluated as iterated (or repeated) 
one-dimensional integrals so that the usual techniques can be used, 
but this doesn't always work.

Does this answer your question adequately, or was there something more 
specific you wanted to know? If you have further questions, please 
write again.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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