Slicing, dicing, and integrating Copyright 2008 by Evans M. Harrell II.Some announcements !!Next test: Thursday the 23rd! Next week!Some announcements From D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie (Geometry and the Imagination)Some announcements From D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie (Geometry and the Imagination)Congratulations to Jacob Schloss! Das Schloß!"#$%&'())#*+,#-,./0/).-,1#23,%.-(,#%4''/5#.&/#6(,7/8#)455'/#%4,#9/#/:;0/) )/5#<-.&#.&/#23,%.-(,! ! " # $ "#" ! "*$=8*">"##?(#2-,5#.&/#-,./0/).-,1#;(-,.)#(2#.&-)#23,%.-(,@#)3%&#4)#.&/#).4.-(,408#;(-,.)#4,5#.&/#1/,/04'#)&4;/@#-.#-)#'-7/ '8#.(#9/#3)/23'#.(#.47/#.&/#1045-/, .#4,5#/A4'34./#)(6/#)46;'/#,(0 64'#A/%.(0)"##?&/#1045-/,.#-)#%4'%3'4./5#.(#9/% ! ! " # $ "#&=:*$=8*'(%)&$B $ "'(&"##?&-)#)&(<)#.&/0/#-)#4#).4.-(,408#;(-,.#4.#CD@D@DE@#4,5#,(< &/0/#/')/"##?(#2-,5#-2#.&/#%/,./0#;(-,.#-)#4#)455'/@#.&/#F/))-4,#64.0-:#%4,#9/#%0/4./5#4,5#4;;'-/5"F/))-4,#G4.0-:H' ! " # $ "#&B: $B8$B8 $B'I#(! " # $"#$=B "$=B8*?&/#5/./06-,4,.#JCD@DE#K#D@#)(#.&-)#%4,,(.#9/#3)/5#.(#2-,5#.&/#;0(;/0.-/)#(2#.&/#(0-1-,"##L,)./45#)(6/#0/;0/)/,.4.-A/#,(064'#A/%.(0)#%4,#9/#;-%7/5#4,5#/:46-,/5"##?&/#,(064'#A/%.(0#-)#*) #% ! ! " # $ # *"#&$=:*)=8*'(%)&B $ " '(&)(+"##+#.49'/#(2#A4'3/)#2(''(<)"M(-,.% ! ! " # $ # * "?4,1/,.#;'4,/!,$,D"+*) #DCD@#D@#DE(+!&"$*'$&DDD'"+*&DD>'#* #DCD@#>@#DE=(%)(+!&"$*'$&D>D'"+*&=D>'#= !"")! *"#DC>@#>@#N*EB(&)(+!&"$*'$&>>$*'"+*&DB>'#B ! $$> ")! *$*"#DC>@#D@#>E$=(%)(+!&"$*'$&>D>'"+*&$=D>'#$=! "$>")! *$>"#DC>@#N>@#N*E$B(&)(+!&"$*'$&>$>$*'"+*&D$B>'#$B! $)>")! *)*"#DCD@#N>@#D@E=(%)(+!&"$*'$&D$>D'"+*&=D>'#=! " ")! * "#DCN>@#N>@#*EB(&)(+!&"$*'$&$>$>*'"+*&DB>'#B !")> ")!*$*"#DCN>@#D@#N>E$=(%)(+!&"$*'$&$>D$>'"+*&D$=>'#$= ! $")! *)>"#DCN>@#>@#*E$B(&)(+!&"$*'$&$>>*'"+*&D$B>'#$B ! $$>")! *$*"#D!"#$%&'())#*+&,#-./0,/-#1'./,#(2#3#4#5#./6#/(78.'#9,%-(7#(2!!#.-#-&,#1(:/-#;5<5<5=#.')(#17(9,#:-#:)#.#)-.-:(/.7>#1(:/-"##?(9:/0#@1#.'(/0#-&,#>#.A:)<#-&,#-./0,/-#1'./,#.-#;5<B<5=#)'(1,)#6(C/C.76)#:/#-&,#1():-:9,#A#6:7,%-:(/#.)#-&,#A#%(81(/,/-#(2#-&,#/(78.'#:)#1():-:9,"##D((E:/0#.-#-&,#/(78.'#9,%-(7)#2(7#-&,#1(:/-)#;B<5<B=#./6#;B<B<FG=<#-&,#>#%(81(/,/-#(2#-&,#/(78.'#:/%7,.),)#.)#(/,#8(9,)#27( 8#./6#-(#-&,#7,)1,%-:9,#1(:/-)<#:/6:%.-:/0#-&,#2@/%-:(/H)#)'(1,#:)#:/%7,.):/0#:/#8.0/:-@6,#:/#-&,#2:7)-#[email protected]./-#.)#A#./6#>#:/%7,.),"##+&,#2@/%-:(/#:)#8(9:/0#6(C/#&:''#J,%.@) ,#-&,#3#9.'@,#(2#-&,#2@/%-:(/#.- #-&,#1(:/-#;B<B<FG=#:)#'(C,7#-&./#-&.-#(2#-&,#(7:0:/<#;B<5<B=<#./6#;5<B<5="##D:E,C:),#2(7#-&,#2(7-&#[email protected]./-<#-&,#/(78.'#.-#;B<FB<FG=#.')(#&.)#.#07,.-,7#>#%(81(/,/-#-&./#-&,#/(78.'#.-#;B<5<B=<#:/6:%.-:/0#-&,#2@/%-:(/#:)#.')(#0,--:/0#)-,,1,7#.)#>#6,%7,.),) #./6#A#:/%7,.),)#27(8#-&,#(7:0:/"##K0.:/<#-&:)#8(9,8,/-#:)#6(C/&:''#J,%.@),#-&,#3#9.'@,#(2#-&,#2@/%-:(/#:)#'(C,7#.-#;B<FB<FG=#-&./#.-#-&,#(7:0:/<#;B<5<B=<#./6#;5<FB<5="?(9:/0#-(#-&,#),%(/6#[email protected]./-<#./6#%(81.7:/0#-&,#/(78.')#2(7#;A< >=#4#;FB<5=<#;FB<B=<#./6#;5<B=<#C,#%./#),,#-&.-#-&,#2@/%-:(/#:/%7,.),)#:/#-&,#6:7,%-:(/#(2"B!"#!#27(8#-&,#(7:0:/<#./6#6,%7,.),)#-(#,:-&,7#):6,#.)#-&,#2@/%-:(/#:)#9.'@,6#&:0&,7#.-#;FB<B=#./6#-&,#/(78.'#)C:/0)#27(8#6(C/&:''#.'(/0#6,%7,.):/0#><#$$ %"L!##!!<#.-#;A<>=#4#;FB<B=#-(#2'.-#:/#>M"*!"#!!.-#;FB<5=#./6*!"#!!.-#;5<B="#+&,#2(7-&#[email protected]./-#):8:'.7'>#&.)#.#1,.E#.-#;FB<FB=#./6#)'(1,)#J.%E#:/#.#2.)&:(/#)>88,-7:%#-(#-&,#-&:76#[email protected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n our previous episode… How do you slice it?In our previous episode… Or dice it? ! "How about some problems that are just like the HW?How about some problems that are just like the HW? 1.! OK – Evaluate the double sumHow about some problems that are just like the HW? 1.! OK – Evaluate the double sum… 2.! Let f(x,y) = x+2y on R = {0!x!2, 0!y!1}. And let us partition the region by P = P1 ! P2. where P1 = {0,1,3/2,2} and P2 = {0,1/2,1}. Find Lf(P) and Uf(P).2. Let f(x,y) = x+2y on R = {0#x#2, 0#y#1}. And let us partition the region by P = P1 ! P2. where P1 = {0,1,3/2,2} and P2 = {0,1/2,1}. Find Lf(P) and Uf(P),2. Let f(x,y) = x+2y on R = {0#x#2, 0#y#1}. And let us partition the region by P = P1 ! P2. where P1 = {0,1,3/2,2} and P2 = {0,1/2,1}. Find Lf(P) and Uf(P),Practicalities of doing double integrals. A double integral ! is ! an iterated integral.!Practicalities of doing double integrals.How about some problems that are just like the HW? 3. EvaluateWhat is the average value of sin(x) sin(y) on that rectangle?It’s time for … !uess "e "eorem! Copyright 2008 by Evans M. Harrell II.What if the integration region is not a rectangle?What if the integration region is not a rectangle? !!Easy cases:What if the integration region is not a rectangle? !!Easy cases: !!Example: 0 ! x ! 2, x ! y ! 2What if the integration region is not a rectangle? !!Easy cases: !!Example: 0 ! x ! 2, x ! y ! 2 !!Example: 1-y ! x ! y ?? ! y ! 4What if the integration region is not a rectangle? !!Not so easy cases:Another game: !!Let’s take one favorite function, like f(x,y) = xy, and integrate it over lots of regions. !!What does the integral of xy over a region in the first quadrant (x,y > 0) represent? !!What if the region is in the second quadrant (x < 0, y > 0)?0 ! x ! 2, x ! y ! 20 ! x ! 2, x ! y ! 2 2 interpretations: a=0, b =2 "1(x)=x#y#"2(x)= 2 Integration path for y at fixed x: from x to 20 ! x ! 2, x ! y ! 2 OR: c=0, d =2 #1(y)=0#x##2(y)= y Integration path for x at fixed y: 0 to y0 ! x ! 2, x ! y ! 21-y ! x ! y ?? ! y ! 41-y ! x ! y ?? ! y ! 41-y ! x ! y ?? ! y ! 41-y ! x ! y ?? ! y ! 4$: between y = 2x + 1 and y = x2. Example$: between y = 2x + 1 and y = x2.$: between y = 2x + 1 and y = x2.$: between y = 2x + 1 and y =