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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 32 "Double Integration and Summatio
n" }}{PARA 0 "" 0 "" {TEXT -1 100 "Goal : Approximate the mass of a pl
ate in the shape of the rectangular region bounded by the curves " }}
{PARA 0 "" 0 "" {TEXT -1 50 "x=0, x=2, y=0, y=1. The density at (x,y) \+
is  xy^2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "restart:\nwith
(plots): \nplate := implicitplot( \{x=0, x=2, y=0, y=1\}, x=0..2, y=0.
.1,\n                title=`find mass of this plate, d(x,y)=1+x+y`,   \+
                                                                      \+
                    color=red, thickness=2, scaling=constrained ):\ndi
splay(plate);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Subdivisions of \+
plate" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "vertlines := seq( x=i/5, \+
i=1..9):\nhorlines := seq( y=j/5, j=1..4):\nmesh := implicitplot( \{ v
ertlines, horlines\}, x=0..2, y=0..1, \n                       color =
 blue, scaling=constrained ):\ndisplay( \{plate, mesh\} );" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "To find mass of
 this plate we make two approximations" }}{PARA 0 "" 0 "" {TEXT -1 45 
"(i) subdivide region into squares with side h" }}{PARA 0 "" 0 "" 
{TEXT -1 146 "(ii) mass of plate  ~= sum of mass of all squares  in th
e plate\nand note \n(iii) mass of each square  ~=   h^2 *  density at \+
mid-pt of this square." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 10 "Density is" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "d := proc(x,y) x*y^2 end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
64 "The approximate mass of the plate for a subdivision of size h is" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "mapprox := proc(h)\n           l
ocal total:\n           total := h^2*sum( sum( d(i*h + h/2,j*h + h/2),
 \n                                    i=0..2/h-1), j=0..1/h-1);\n    \+
       evalf(total);\n           end:  " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 50 "So the approximate mass of the plate when h=0.1 is" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "mapprox(0.5);" }}}{PARA 0 "" 0 "" 
{TEXT -1 80 "Decrease h from 0.1 to 0.01 - what is the approximate mas
s. Decrease it further." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 18 "The exact masss is" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "M := Int( Int( d(x,y), x=0..2 ), y=0..1 );\nevalf(M);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "2 0" 49 }
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