Change of Variables. f(x, y) da = (1) If the transformation T hasn t already been given, come up with the transformation to use.

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1 MATH 2Q Spring 26 Daid Nichols Change of Variables Change of ariables in mltiple integrals is complicated, bt it can be broken down into steps as follows. The starting point is a doble integral in & y. f(, y) da () If the transformation T hasn t already been gien, come p with the transformation to se. (2) Figre ot both directions of the transformation. (3) Fill in the following information in any order: f(, y) da = S f((, ), y(, )) (, ) d d (a) change to S Note that depending on the problem yo might se either order of integration, d d or d d. Eample Use the transformation = 4 ( + ), y = 4 ( 3) to ealate the integral (4 + 8y) da, where is the parallelogram with ertices (, 3), (, 3), (3, ), and (, 5). I hae already been gien the transformation. 2 To get & in terms of & y, I sole for and in the system I was gien (notice that I e distribted ot the 4 in each eqation): = () y = (2) Sbtracting (2) from () gies me = y, and adding three times () to (2) gies me = 3 + y. 3(a) This is nearly always the hardest step, and it can reqire a lot of work. In order to figre ot S, I first need to nderstand what looks like. So I m going to draw a pictre. It isn t strictly necessary to draw a pictre, bt I find that if I hae a pictre to work with, the rest of the problem gets easier. The problem statement has told me that is a parallelogram with certain corners, so the first step to draw is to mark the corner points I was gien. The reslt appears on the net page. We need to know how to get (, y) from (, ) to calclate the Jacobian, and we need to know how to get (, ) from (, y) to figre ot the region of integration.

2 y (, 5) (, 3) (3, ) (, 3) Then I connect the dots to draw : y (, 5) (, 3) (3, ) (, 3) The point of drawing this pictre of (the region of integration on the, y side of the transformation) was so that I cold figre ot S (the region of integration oer on the, side of the transformation). To figre ot S, all I hae to do is figre ot the bondary and then eerything else will fill itself in. In fact, all I hae to is figre ot either (i) what the transformation does to each of the 4 sides of, or else (ii) what the transformation does to each of the 4 corners of. Once I do either of those, the rest of S will fill itself in. I know from step (2) that = y, = 3 + y. So plgging in the for corner points of gies me the corners of S: (, y) (, ) Corner : (, 3) (, ) Corner 2: (, 3) (4, ) Corner 3: (3, ) (4, 8) Corner 4: (, 5) (, 8)

3 Now I can mark the corners of S... (, 8) (4, 8) (, ) (4, )...and fill in the rest. (, 8) (4, 8) S (, ) (4, ) I e now got the hardest part ot of the way, and I can fill in part of the integral: (4 + 8y) da = f((, ), y(, )) (, ) d d (a) change to S 3(b) Since I already know what & y are as fnctions of &, I can jst plg in that information: [ ] [ ] 4 + 8y = 4[(, )] + 8[y(, )] = 4 ( + ) + 8 ( 3) =

4 Now I can fill in another part of the integral: (4 + 8y) da = (3 5) (, ) d d (a) change to S 3(c) The last ingredient I need for the change of ariables is the Jacobian. Since I know what & y are as fnctions of &, I can take the deriaties and plg them in to get the Jacobian: [ ] [ ] (, ) = det /4 /4 = det = 3/4 / = 4. This is the last part of the integral that I need to fill in: (4 + 8y) da = (3 5) 4 d d (a) change to S Now that I m done with the change of ariables, I can ealate the integral in the sal way: (4 + 8y) da = (3 5) 4 d d = 4 = 4 = 4 = (3 5) d d (factor ot the /4) [ ] =8 d = (inside integral first) (96 4) d (plg in limits) [ ] 4 (now the integral) = 92 (plg in limits)

5 Eample 2 Make an appropriate change of ariables to ealate the integral ( + y)e2 y 2 da, where is the rectangle enclosed by the lines y =, y = 2, + y =, and + y = 3. I need to come p with a change of ariables that cold make this integral easier. There are two ways I cold approach this: I cold try to make the integrand nicer or I cold try to make the region of integration nicer. Alternatiely, I cold try to look at both the integrand and the region of integration while I try to come p with the transformation to se. After staring at the problem for a while, I start to notice some patterns. ( + y) shows p in a lot of places: it s mentioned twice in the description of, it appears in the integrand, and in fact if I write ( + y)e 2 y 2 = ( + y)e (+y)( y) then I notice that ( + y) shows p twice in the integrand. So I cold simplify a lot of pieces of the problem down to jst if I set = + y. That leaes me with e ( y) in the integral, so the natral choice of is = y, which trns the integrand into e. 2 I already know how to find & in terms of & y, so now I want to know how to find & y in terms of &. In other words, I need to sole this system for and y: Adding ()+(2) gies me + = 2, so = + y () = y (2) = ( + ), 2 and sbtracting () (2) gies me = 2y, so y = ( ). 2 3(a) In order to find S, I first need to nderstand. Once again, I ll draw a pictre since that s the easiest way for me to analyze the information that I hae. The problem describes based on 4 lines, so the first thing I m going to do to make a pictre of is draw those lines: y + y = 3 y = y = 2 + y =

6 The problem statement tells me that is the rectangle enclosed by those lines: y y = + y = 3 + y = y = 2 Since I know how to trn & y into &, I can now apply the transformation to each of the sides of in order to get the sides of S. Now I can draw the sides of S..., y, y = = y = 2 = 2 + y = = + y = 3 = 3 = 2 = = = 3...and fill in the rest. S With the hardest part ot of the way, and I can fill in part of the integral: ( + y)e 2 y 2 da = f((, ), y(, )) (, ) d d (a) change to S

7 3(b) I already saw how the integrand is transformed back when I was coming p with the transformation to se: ( + y)e 2 y 2 = ( + y)e 2 y 2 = ( + y)e (+y)( y) = e. Ths ( + y)e 2 y 2 da = e (, ) d d (a) change to S 3(c) I know what & y are as fnctions of &, so I can take the deriaties and plg them in to get the Jacobian: [ ] [ ] (, ) = det /2 /2 = det = /2 /2 4 4 = 2. This is the last part of the integral that I need to fill in: ( + y)e 2 y 2 da = e 2 d d (a) change to S Now that I m done with the change of ariables, I can ealate the integral in the sal way: 2 3 ( + y)e 2 y 2 da = e 2 d d = 2 = e d d [e ] =2 d = = (e 2 ) d 2 = [ ] e2 = ( 2 2 e = e6 7 4 )