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How to Integrate a Polynomial over a Sphere Author(s): Gerald B. Folland Source: The American Mathematical Monthly, Vol. 108, No. 5 (May, 2001), pp. 446-448 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2695802 Accessed: 03-06-2015 16:13 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/2695802?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 165.91.112.105 on Wed, 03 Jun 2015 16:13:03 UTC All use subject to JSTOR Terms and Conditions REFERENCE 1. I. Panakis,Plane Trigonometry,(in Greek) Vol. II, privatepublication,Athens, 1973. 81 PatmouStreet,Athens, 11144, Greece [email protected] Florida Atlantic University,Boca Raton, FL 33431 [email protected] How to Integrate A Polynomial Over A Sphere Gerald B. Folland Severalrecent articlesin the MONTHLY ([1], [2], [4]) have involvedfindingthe area of n-dimensionalballs or spheres or integratingpolynomialsover such sets. None of these articles,however,makesuse of the most elegantand painlessmethod for performingsuch calculations,which is to reversethe usual trick for computing 2 flOe dx. Thereis nothingnew in this idea. It was shownto me in 1971 by V. Bargmannand E. Nelson,andI includedit as an exercisein my book [3], whose firsteditionappeared in 1984.But the evidencesuggeststhatit is not as universallyknownas it shouldbe. First,some notation:For x = (xi, ... xn) E Rn, we set Ix = (X2+ * + x2)1/2, and we denote the sphereand ball of radiusr aboutthe origin in Rnby Sn(r) and Bn(r), respectively: , Sn(r)= {x E Rn: lxl = r}, Bn(r)= {x E Rn: lxI < r}. We also set Sn = Sn(1), Bn = Bn(l). Any nonzerox E Rn can be writtenuniquelyin "polarcoordinates": x = rx', x where r = lxI and x' =-E Sn. lxl Let a denotethe (n - l)-dimensional surfacemeasureon Sn;thenthe formulafor integrationin polarcoordinatesis f f(x)dx = f(rx') rn-1dr da(x'). f (1) (A briefderivationis sketchedin Remark2.) Ourobjectis to computefS P du where P is a polynomial,andfor this it sufficesto considerthe case whereP is a monomial, P(X) 446 ? = X = xx22. 2 .. n (al, aXn E {O,1 2,...}) (2) THE MATHEMATICALASSOCIATION OF AMERICA [Monthly 108 This content downloaded from 165.91.112.105 on Wed, 03 Jun 2015 16:13:03 UTC All use subject to JSTOR Terms and Conditions The answeris expressedin termsof the gammafunction, (t) = J e-s stl ds = 2 J dr. r2t1e-r2 The relevantvaluesof t are integersor half-integers;we recall that F(k) = (k - 1)! and F(k + 2) = (k - 2) when k is an integer. ) *** Theorem. Let P be given by (2), and let f3j = 2 (aj + 1). Then if some aj is odd, I 0 2FP(j)F(/2)...P(/On) +/32+ *-+ P) = ]Pd Jr(V1 if all aj areeven. Proof. If some aj is odd,.the integralvanishesby symmetry.Assume thatall oj are even, andconsiderthe integral I= P(x)e-x12 dx. j Evaluationof I in Cartesiancoordinatesgives 0, o n n I=H]I } xi e- dxj = ,00 2, On the other hand, since P(rx') = e-xj dxj = F(/1) xJ ral+ +nnP(x'), ( ..) .(rfn). (3) evaluationin polar coordinates gives p 00 P(rx')e f rrn- dr da(x') raul+ +aUn+n-1 er2 + + PO) r(/1 2 dr P (x') du c(x') p (x') d c(x'). (4) s Comparisonof (3) and(4) yields the desiredresult. U Theintegralof a polynomialP overa ball is nowreadilyobtainedby takingf (x) = P(x)x (x) in (1), where x is the characteristicfunctionof the ball. Namely,if P is givenby (2), R P(x) dx ] Bn(R) = rul+ +Un+n-l dr JOl In particular,for the case P +fn+n rRol Pda = + + n+ n / P d. 1, we have: Corollary. The (n - 1)-dimensional area of Sn is 2wTn/2/r (n/2), and the n-dimensional volume of Bn is 2r n2/n2 (n/2). May 2001] NOTIES This content downloaded from 165.91.112.105 on Wed, 03 Jun 2015 16:13:03 UTC All use subject to JSTOR Terms and Conditions 447 Remark 1. The samecalculationsyield the integralof IxiI1"... Ix n over a sphere nonnegativerealnumbers.Thisgeneralizationwas or a ball whenthe aj's arearbitrary also consideredin [2]. Remark2. Here is a sketchof the constructionof the surfacemeasurea and the proofof (1). The startingpointis the fact thatdilationby a factorof r > 0 multiplies if c is the volumeof Bn(1), then n-dimensionalvolumesby a factorof r . In particular, the volumeof Bn(r) is cr . It follows thatthe volumeof a sphericalshell of radiusr andinfinitesimalthicknessdr is d(crn) = ncr n-1 dr. On the otherhand,this volume is the productof the thicknessdr andthe surfaceareaof the sphereSn(r), so thatarea must be ncr-1. The samereasoningappliesto a sectorratherthana whole ball. Thatis, considera subsetU of Sn= Sn(1) andthe correspondingsectorQ?u(r)of Bn(r), Qu(r) = {sx': 0 < s < r, x' E U} . Then if curn is the volume of Qu (r), the surface area of U must be (d/dr) (curn)lr1 = ncU. We are therefore led to define surface measure a on Snby a (U) = n *vol(Qu (1)). of the radianmeasureof an angle thatis used (Forn = 2, this is the characterization in the standardgeometricproofthatlim,0 (sin x)/x = 1.) Finally,if we take U to be an infinitesimalbit of Snwith measureda, we see thatthe volumeof the infinitesimal box {sx' r <s < r + dr, x' E U} is rn-1 dr da, from which (1) follows. To see these calculationsfully clothedin rigorousmeasuretheory,the readermay consult [3, Theorem2.49], where (1) is proved for Lebesgue integrablefunctions on Rn. A moreelementarytreatmentof the case of continuousfunctionscan be found in [2]. REFERENCES 1. L. Badger, Generatingthe measure of n-balls, this MONTHLY 107 (2000) 256-258. 2. J. A. Baker, Integrationover spheres and the divergence theorem for balls, this MONTHLY 104 (1997) 36-47. 3. G. B. Folland, Real Analysis (2nd ed.), John Wiley, New York, 1999. 4. 0. Hijab, The volume of the unit ball in Cn, this MONTHLY 107 (2000) 259. Universityof Washington,Seattle, WA98195-4350 [email protected] 448 ( THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108 This content downloaded from 165.91.112.105 on Wed, 03 Jun 2015 16:13:03 UTC All use subject to JSTOR Terms and Conditions