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Topic: Double integral

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	Double Integrals: Defined (Site not responding. Last check: 2007-11-06)
		The integral of a function f over some set D (included in the domain of definition of f) is a limit and we say that the integral exists if this limit exists.
		The limit that defines an integral is of the same general form for all integrals no matter if the function is of one variable (as in calc1) or of several variables (as here in calc3).
		In this way the computation of a double integral is reduced to computing two one dimensional (just as in calc1) integrals one after the other.
	omega.albany.edu:8008 /calc3/double-integrals-dir/define-m2h.html (928 words)

	The Integral Document (Site not responding. Last check: 2007-11-06)
		An integral is applicable to a continuous function of an interval on a definite variable of the function.
		Definite integrals are used to calculate, in general, numeric values of areas, volumes, centroids, lengths, etc.
		In the illustration a and b are the limits of integration, the color lines are area elements, dx the lenght of the area elements and f(x) the height of the area element.
	www.area48.com /integral/index.html (910 words)

	Double integral - Wikipedia, the free encyclopedia
		In mathematical analysis, there is an important distinction between a double integral and an iterated integral.
		and in that case, the double integral coincides in value with either of the two iterated integrals.
		Thus, whenever the two iterated integrals differ in value from each other, the double integral of the absolute value of the function must be infinite.
	en.wikipedia.org /wiki/Double_integral (347 words)

	Multiple integral - Wikipedia, the free encyclopedia
		This is an instance of rearrangement of a conditionally convergent integral.
		In mechanics the moment of inertia is calculated as a volume integral (that is a triple integral) of the density weighed with the square of the distance from the axis:
		In case of unbounded domains or unbounded integrands near the frontier of the dominion you have the double improper integral or the triple improper integral.
	en.wikipedia.org /wiki/Multiple_integral (2396 words)

	Double integrals*
		The integral is the limit of the Riemann sums as the size of the rectangles goes to zero.
		We have better ways to compute double integrals (that is, unless you are a computer, in which case chopping up the domain in pieces and computing a sum as an approximation to an integral works pretty well).
		interpret the double integral as volume underneath a surface, just like you could interpret the regular one-variable integral as area under a curve.
	www.math.umn.edu /~nykamp/m2374/readings/doubleint/index.html (695 words)

	Evaluating Double Integrals
		Evaluating a multiple integral involves expressing it as an iterated integral, which can then be evaluated either symbolically or numerically.
		We will now address the problem of determining limits for a double integral from a geometric description of the region of integration.
		Since the disk in which we are interested lies entirely to the right of the x-axis, the appropriate limits for theta are -pi/2 and pi/2.
	www.math.umd.edu /~jmr/241/doubleint.html (1240 words)

	Double Integrals and Volume
		In first year calculus, the definite integral was defined as a Riemann sum that gave the area under a curve.
		Notice that all the typical properties of the double integral hold.
		For example, constants can be pulled out and the double integral of the sum of two functions is the sum of the double integrals of each function.
	ltcconline.net /greenl/courses/202/multipleIntegration/Volume.htm (513 words)

	[No title]
		The limit process in a double integral occurs simultaneously in both variables.
		In an iterated integral the integr ation (limit) occurs in one variable and then the other, so it is two \+ step process.
		Student difficulties with double integrals center aroun d setting up the limits of the equivalent iterated integral.
	www.runet.edu /~wyang/252/DoubleIntegral.mws (447 words)

	A Flux Integral Applet
		To evaluate the integral we want to convert it to a simple double integral of a real valued function over a region in s and t.
		It reduces the flux integral to the double integral of a real valued function over a region in s and t.
		The default vector field is F = (x, z, y) and the default surface is an eighth of the unit sphere.
	www.slu.edu /classes/maymk/banchoff/FluxIntegral.html (412 words)

	Integration Over General Regions (Site not responding. Last check: 2007-11-06)
		Notice, that if f(x,y) is non negative the double integral over D can still be interpreted as the volume that lies between D and the surface z=f(x,y).
		Consider the region D between the circles of radius 1 and 2 in the first quadrant.
		write the double integral of a general function g(x,y) over the region D as itereted integrals over x and y.
	omega.albany.edu:8008 /calc3/double-integrals-dir/gen-region-m2h.html (281 words)

	Calculus III (Math 2415) - Multiple Integrals - Double Integrals in Polar Coordinates
		Due to the limits on the inner integral this is liable to be an unpleasant integral to compute.
		We are now ready to write down a formula for the double integral in terms of polar coordinates.
		First, notice that we can not do this integral in Cartesian coordinates and so converting to polar coordinates may be the only option we have for actually doing the integral. Notice that the function will convert to polar coordinates nicely and so shouldn’t be a problem.
	tutorial.math.lamar.edu /AllBrowsers/2415/DIPolarCoords.asp (1583 words)

	14.2
		The exact answer may be obtained from a double integral.
		The first step in evaluating a double integral is to rewrite it as an iterated integral.
		The average value of the function f over the region R can be calculated by dividing the double integral of the function over the region by the area of the region.
	www.ac.cc.md.us /~donr/CalcIII/unit4/lesson2/u4l2.html (794 words)

	: Class NumAnal
		integral of the static function named fcn in Class cls from a to b.
		integral of the function named fcn in Object obj from a to b.
		When only one of the limits is infinite, AA is set to the finite endpoint and BB is made negative or positive as appropriate.
	www.cs.uiowa.edu /~rlenth/Power/doc/rvl/util/NumAnal.html (484 words)

	Double Integral Sanitation Ltd. (Site not responding. Last check: 2007-11-06)
		Double Integral was incorporated 24 years ago as a complete
		Double Integral Sanitation originated because of an outbreak
		Double Integral Sanitation Ltd. was incorporated in 1984
	www.doubleintegral.ca (245 words)

	Numeric calculus for double integrals
		A second reason that greatly complicates the multidimensional integration is the boundary that for a 2D Integral is a region of x-y plane.
		By contrast, the boundary of one-dimensional integral consists of two constant - upper and lower limits.
		For each of one we have reported the values returned by the routine: the approximated integral, the estimated relative error, and the number of evaluated nodes.
	digilander.libero.it /foxes/integr/double_integrals.htm (1085 words)

	NMath Core User's Guide - 10.2 Numerical Integration
		TrapeziodEstimate gets the estimate for the integral yielded by the compound trapeziod rule where the number of panels is equal to the order of the Romberg estimate.
		SimpsonEstimate gets the estimate for the integral yielded by the compound Simpson's rule where the number of panels is equal to the order of the Romberg estimate.
		GaussKronrod43Integrator approximates integrals using the Gauss 21-point and the Kronrod 43-point rule.
	www.centerspace.net /doc/NMath/Core/user/calculus3.html (1394 words)

	Double Integrals (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-06)
		If we begin by declaring x and y to be symbolic variables, we may omit the quotation marks; you will probably want to make use of this most of the time.
		numint2, which have identical calling syntax and implement, respectively, symbolic double integration using int, and numerical double integration using dblquad.
		Since the disk in which we are interested lies entirely to the right of the x-axis, the appropriate limits for theta are -p/2 and p/2.
	www.math.umd.edu.cob-web.org:8888 /~jmr/241/doubleint.htm (1037 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		otherwise you may triangularize the region and use special formulae for triangles (reducing them to the standard triangle) codes for trinagularization and two-dimensionla integrals are in netlib/toms.
		in stroud: approximate calculation of multiple integrals, prentice hall 1971 or in engels : numerical quadrature and cubature, academic press 1980.
		These rumours, which seem reasonable to > me, are to effect that numerical evaluation of a double integral is > riskier to perform than the evaluation of a single integral.
	www.math.niu.edu /~rusin/known-math/99/double_int (794 words)

	double integral sign - a definition from Whatis.com
		An example is the volume under a surface in three-dimensional (3D) space.
		The double integral sign appears as two single integral signs, placed one after the other:
		A typical expression for an indefinite integral of a function f of two variables x and y using the double integral sign is:
	whatis.techtarget.com /definition/0,,sid9_gci874054,00.html (149 words)

	Computing Double Integrals with ...
		When we try to have Maple compute this integral, we have to wait a long time to find out that Maple can't do it.
		There are times when reversing the order of integration will not enable you or Maple to solve a particular double integral problem.
		Note if we use the output = integral option -- the integral is shown unevaluated (this is called the inert form in Maple) -- to evaluate we would then use the value command.
	www.saintjoe.edu /~karend/m236/DoubleIntegrals.mw (453 words)

	Cephes double precision special functions suite
		However, * the low order bits of a double precision number may * not be adequately random, and may vary due to arithmetic * implementation details on different computers.
		(b) - * 0 * * The incomplete gamma integral is used, according to the * relation * * y = igam(b, ax).
		* The integral from x to 1 may be obtained by the symmetry * relation * * 1 - incbet(a, b, x) = incbet(b, a, 1-x).
	www.netlib.org /cephes/doubldoc.html (16786 words)

	Symmetrical double integral
		The two-sided leaky integral commonly arises as an even function, which is an ordinary leaky integral in one direction followed by another in the opposite direction.
		We will see also that the single leaky integral need not be causal; it could be an odd function.
		It is convenient to refer to the symmetrical double integration operator as
	sep.stanford.edu /sep/prof/pvi/zp/paper_html/node11.html (403 words)

	How to code the double integral ? \| MaplePrimes (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-06)
		since the qx(x)'s is not keep constant on whole range, I am confused with the double integral.
		This is pretty straight-forward except that Maple does not do well if the full integral is entered all in one command.
		The problem is that the bounds on a definite integral are not automatically applied as ``assumptions'' when the integrand is being evaluated.
	beta.mapleprimes.com.cob-web.org:8888 /forum/how-to-code-the-double-integral (268 words)

	double integration (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-06)
		This tutorial starts with a definition of an integral in one variable, and then extends the definition to cover integrals over multiple variables.
		It breaks down the discussion of the double integral into different sections, allowing for a greater overall clarity of the site.
		There is a link near the top of the page to go to triple integrals in cylindrical and spherical coordinates.
	www.math.umd.edu.cob-web.org:8888 /undergraduate/resources/TutorialPages/multiple_integration.html (277 words)

	135: Create a closed double integral symbol (Site not responding. Last check: 2007-11-06)
		These instructions explain how to create the closed double integral.
		In the Operator dialog box, select the double integral.
		The closed double integral should appear correctly in the previewer.
	www.mackichan.com /techtalk/how/howdoubleintcirc.html (121 words)

	Double Integral - Math Help Forum
		Then set the interval for the outer integral, say x from 0 to the x-intercept.
		Then the interval for the inner integral is from y=0 to y as a function of x.
		Then evaluate the partial integrals starting from the inner integral for y.
	www.mathhelpforum.com /math-help/calculus/43-double-integral.html (458 words)

	Double Integral -- from Wolfram MathWorld
		A double integral is a two-fold multiple integral.
		Examples of definite double integrals evaluating to simple constants include
		Triple Integral, Unit Disk Integral, Unit Square Integral.
	mathworld.wolfram.com /DoubleIntegral.html (92 words)

	Double Integral
		I need to solve a double integral and I have no idea what to change the variables to:
		After putting in these variables, I can't solve the integral.
		If I change what u and v are, then either the Jacobian is zero or I still can't solve the intergral.
	www.physicsforums.com /showthread.php?p=828913 (655 words)

	need help with double integral problem - NASIOC
		//sin x² dx dy with the integrals being 0,1 and y,1.
		start out by doing the inside integral first (sin x^2 dx) then apply the boudaries y,1.
		That's a single variate integral, since y isn't even in your function.
	forums.nasioc.com /forums/showthread.php?t=558683 (410 words)

	2006 Integral Yoga Double Anniversary Calendar (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-06)
		The newly remastered DVD of this extremely popular Level I Hatha class taught by Sri Gurudev is NOW AVAILABLE
		"The goal of Integral Yoga, and the birthright of every individual, is to realize the spiritual unity behind all the diversities in the entire creation and to live harmoniously as members of one universal family.
		Notify me of updates to 2006 Integral Yoga Double Anniversary Calendar
	www.shakticom.org.cob-web.org:8888 /product_info.php?products_id=323 (215 words)

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