Integrals versus AntiderivativesDate: 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/ |
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