A Tutoial on Multiple Integals (fo Natual Sciences / Compute Sciences Tipos Pat IA Maths) Coections to D Ian Rud (http://people.ds.cam.ac.uk/ia/contact.html) please. This tutoial gives some bief eamples of whee multiple integals aise in science, and then tackles the issue which causes most confusion fo students: how to wok out the limits on the integals when integating ove a given egion.. Some Eamples (a) The multiple integal d d dz ove some egion of thee dimensional space R R epesents the volume of the egion R. In this eample, the integand (the function we ae integating) is equal to one. (b) If the integand is not simpl one, then we might get something like,, z d d dz ρ,, z is the mass densit at an point within the egion R ρ ( ). If ( ) R, then the multiple integal epesents the total mass within the egion R. In effect, one is multipling densit b volume to get the mass, but allowing fo the fact that the densit might va ove the egion. If ρ (,, z) is the chage densit athe than the mass densit, then the multiple integal epesents the total chage contained within the egion. R (c) The multiple integal ( ) suface ( ) z, d d can be egaded as the volume unde a z, within a two dimensional egion of the -plane R.. How to Find the Limits The pocess that students most stuggle with at fist is how to find the limits on the integals, given a egion. Thee eists a sstematic method fo doing this, and if ou follow it, ou should get the coect answe; if ou do not follow it, ou ma well not get the coect answe. We will take a specific eample to demonstate. Imagine we wish to find the volume unde the suface z + within the egion shown below:
-/ Figue We can imagine making up the volume fom a set of volume elements, each of height z + and base dimensions and. Viewed fom above, one element would look like this: and fom the side: z+ Figue The volume would then be the sum of the volumes of the elements needed to cove the entie egion: ( + ) We can ca out both summations using integation, b allowing and to tend to zeo. But we have to be caeful of the limits on the integals. Let's assume we choose to do the -sum (ie -integal) fist. We'll also do it -fist late on. Imagine that we eplicate the element so that we make a line of elements, stetching in the -diection:
(- ) Figue Ask ouself: what is the minimum of these elements, and what is the maimum of these elements? The minimum is and the maimum is ( ). Note that the maimum is not : it depends on whee the line of elements is on the -ais. This pocedue gives us the limits fo the integal in. To find the -limits, we now imagine ou single line of elements above being eplicated to cove the entie egion: Figue 4 Ask ouself: what is the minimum of these lines of elements, and what is the maimum of these lines of elements? The minimum is and the maimum is. This gives us the limits fo the integal in. Hence the oveall integal is: ( ) ( + ) d d When evaluating the integal, we do the -integal fist. When eading an epession like the one above, do not be fooled into thinking that the -integal is done fist because its integal sign comes fist. We do the inne integal fist, and then the oute integal. While we ae doing the -integal, we egad as a constant, which is consistent with Figue. So we get:
( ) [ / + ] ( ) + ( ) ( ) d [ ] d d [ ] If instead we chose to do the -sum (ie the -integal) fist, then the pocedue fo finding the limits consists of imagining a line of elements along fo some fied : Figue 5 That would give us a minimum of and a maimum of /. We would then eplicate the lines of elements to make it cove the egion: Figue 6 The minimum is and the maimum is. So the integal is:
( + ) / d d [ + / ] / d [ / ] ( / ) + ( / ) d 8 + + d 8 + 4 +. The two outes (-fist and -fist) must give the same answe. Sometimes one is quicke than the othe, and occasionall one is simpl not feasible when the othe is, so ou ma need to choose the ode of integation with cae.. Multiple Integals in non-catesian Coodinate Sstems Although the eamples above all use the Catesian coodinate sstem, it is common to switch to a diffeent coodinate sstem to do multiple integals. The eason fo this is that the integals ma be difficult in Catesians. Fo eample, if one wishes to integate + ove a unit cicle, following the pocedue above, the limits would be to + and then - to +. Afte substituting in the - limits, one is left with an unpleasant integal in. B contast, if one does the integal in plane polas, then the limits ae much simple. One can still use the conceptual pocess descibed above. Fistl, imagine a single plane pola element within the cicle: Figue 7 Then eplicate the element, fistl in the -dimension to poduce a line of elements, and then in the -dimension to cove the whole cicle. The -limits ae simpl to
and the -limits ae to. We can eplace the integand + with. But what do we eplace d d with? A igoous answe to this question tuns out to be complicated, but a simple pocedue will give us the answe. In Figue 7, one can see that the sides of the element ae of length and, and if and ae ve small then the element is appoimatel ectangula. So its aea is, o. It tuns out that if we simpl eplace d d with dd, we get the coect integal: d d d d At this point we could do the integal, followed b the integal, o we can speed things up a little b noticing that the limits ae constants. This allows us to split the double integal into a poduct of one dimensional integals, and pocess them in paallel: d d [ ] 4 4