integral over plane region
The integrals over a planar region are generalisations of usual Riemann integrals, but special cases of http://planetmath.org/node/6660surface integrals.
0.1 Integral over a rectangle
Let be the rectangle of -plane defined by
and the function be defined and bounded in . Let
a of into the rectangular parts with areas (). Denote
Let then be defined in a region of -plane such that that it can be enclosed in a rectangle defined by (1). Define the new function through
Definition 2. If is integrable over the rectangle , we say that is integrable over and define
It’s apparent that (4) is on the choice of since the points of give zero-terms to the lower and upper sums.
0.2 Double integrals
Definition 3. Let be bounded in as before. Suppose that
is defined on . If also the integral
exists, it is called a double integral or iterated integral and denoted by
One may prove the
Theorem. , provided that the integral of the left side exists and that the inner integral of the right side exists for every in .
It’s clear that if also the integral exists for every in . If especially the function is continuous in the rectangle , then surely
Assume now, that is defined and bounded in the region
where is contained in the rectangle determined by (1). Then the planar integral is defined as
if the integral of right side exists. For this, he continuity of in does not necessarily suffice, because may have a jump discontinuity on the border of whence the integrability of needs not be guaranteed. One case where the integrability is true is that the graphs of the functions and are rectifiable (i.e. the functions have continuous derivatives). For a continuous , we then have
i.e. the planar integral has been expressed as a double integral.
[Not ready …]
|Title||integral over plane region|
|Date of creation||2013-03-22 18:50:57|
|Last modified on||2013-03-22 18:50:57|
|Last modified by||pahio (2872)|