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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT -1 13 "MAT 156 sp'08" }}{PARA 
208 "" 0 "" {TEXT -1 5 "Lab 3" }}{PARA 207 "" 0 "" {TEXT -1 34 "More o
n Riemann Sums and Integrals" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "You should not do this l
ab until you have completed Lab 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 53 "You can add color to your leftbox comman
d as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(stude
nt): with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "leftbo
x(x^2,x=0..4,5,shading=yellow);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Now I am going to combine two left
sum pictures with different colors." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "lb10:=leftbox(x^2,x=0..4,10,shading=yellow):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lb5:=leftbox(x^2,x=0..4,5,sh
ading=white):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(lb
5,lb10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 344 "You can easily see the difference between the two left s
ums now.  (Warning:  Maple seems tempermental about these commands.  F
or example writing display(lb10,lb5) instead of display(lb5,lb10) matt
ers.  So be patient and experiment wtih the commands.)  The problem is
 that the first plot covers the second and the second does not show th
rough.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 206 20 "Assignment Part 1:  " }{TEXT -1 20 " (Do not forget the \+
" }{TEXT 256 5 "plots" }{TEXT -1 6 "  and " }{TEXT 257 8 "student " }
{TEXT -1 43 "packages).   Using sin(x) and x=0..Pi/4.   " }}{PARA 0 "
" 0 "" {TEXT -1 117 "A)    Graph the left-hands sums with 5 and 10 sub
intervals simultaneously. Explain why LHS(5) is less than LHS(10).  " 
}}{PARA 0 "" 0 "" {TEXT -1 52 "B)    Graph and explain why RHS(5)>RHS(
10)>RHS(20). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 207 20 "Assignment Part 2:  " }{TEXT -1 27 "Using sin(x) and \+
x=0..Pi/2." }}{PARA 0 "" 0 "" {TEXT -1 108 " Write commands to compute
 the left-hand sums and the right-hand sums with 10, 20, 40, 80, 160 s
ubintervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 410 "We see that indeed the left-hand sums are increasing a
nd the right-hand sums are decreasing. At the same time the right-hand
 sums are always larger than all the left-hand sums.\nAs you see we ha
ve not gotten great accuracy.   To compute more accurately it will be \+
nice to increase the\nnumber of subintervals. As it is tiring to write
 the same commands over and over, it is important to introduce loops i
n Maple." }}}{EXCHG {PARA 203 "" 0 "" {TEXT -1 1 "\n" }{TEXT 208 39 "L
oops and convergence of Riemann sums.\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 419 "The following commands produce a table of the left-hand \+
sums  with 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120 subinter
vals. Notice that every time we multiply the number of subintervals by
 2, so at the first step (j=0) we have 5*2^0 subintervals, at the seco
nd step (j=1) we have 5*2^1=10 subintervals, at the third step (j=2) w
e have 5*2^2=20 subintervals etc. This gives the formula 5*2^\{j\} in \+
the commands. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "for j from 0 to 10 do n:= 5*
2^(j): evalf(leftsum(x^2, x=0..4, n)):od;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "We see tha
t the first two decimal places have stabilized. However, as you have n
oticed before, the left-hand sums are increasing. To be certain that t
he second decimal place is indeed 2 we need to give overestimates, whi
ch are the right-hand sums. " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }
{TEXT 209 19 "Assignment Part 3: " }{TEXT -1 197 "  Write commands tha
t produce the right-hand sums with the same number of subintervals, us
ing a loop. All you have to do is copy the command for the left-hand s
ums and change leftsum into rightsum." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 200 30 "Computing Integrals with M
aple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(x^2,x=0..4); " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 19 "Assignment Part 4: " }{TEXT 
-1 9 "  Study  " }{XPPEDIT 18 0 "int(1/x, x = 1 .. 2);" "6#-%$intG6$*&
\"\"\"F'%\"xG!\"\"/F(;F'\"\"#" }{TEXT -1 12 " as follows:" }}{PARA 0 "
" 0 "" {TEXT 211 3 "A) " }{TEXT -1 120 " Compute the left-hand sums an
d right-hand sums with 2, 4, 8, 16, 32, 64, 128, 256, 516, 1024 subint
ervals. Use a loop. " }}{PARA 0 "" 0 "" {TEXT 212 2 "B)" }{TEXT -1 46 
"  How many decimal digits are you certain of?\n" }{TEXT 213 3 "C) " }
{TEXT -1 64 "  Which are overestimates and which are underestimates an
d why? " }}{PARA 0 "" 0 "" {TEXT 214 3 "D) " }{TEXT -1 150 "  Do the l
eft-hand sums form an increasing or decreasing sequence of numbers, as
 the number of subintervals increase? Show this in appropriate graphs.
" }}{PARA 0 "" 0 "" {TEXT 215 3 "E) " }{TEXT -1 29 " Compute the exact
 integral.\n" }}}}{MARK "10 1 0" 61 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }