Using Riemann Sums to Compute and Approximate Definite Integrals In this tutorial, we demonstrate how Maple is useful in computing Riemann sums. We start by using the restart command to make sure all variables are unassigned. restart;

Sometimes for memory purposes not all the built-in Maple functions are available for use in a standard Maple. However, there are many packages of functions designed for special purposes that are available and can be read into the session by enclosing the package name within the with statement. Some of the functions we will need are contained in the student package. Before these routines can be used, these commands must be read in into the session by entering the following command. with(student);

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The output of the command lists the routines that are now available for use in the Maple session.

Suppose we want to approximate the area under the graph of 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 which is the same as approximating the definite integral 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. The next command enters 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 as a Maple function. f := x -> x^2 + 1;

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In this session, we do our approximations using left endpoints, midpoints, and right endpoints. To see how the rectangles are drawn using left endpoints, we use the leftbox command. In the the leftbox command, we will use the syntax leftbox(function, x-axis interval, number of rectangles, x and y axis view) where function is the function to be used, x-axis interval gives the endpoints of the definite integral displayed in the the syntax x = lowerend..upperend, and number of rectangles is the number of rectangles (subintervals) to be used. The x and y axis view is an optional parameter involving the view plot option which allows one to specify how much of the x and y axis to display on the graph. As an example, the next command will graph the rectangles using left endpoints using the function f(x) stored above on the interval x = 0..2 using 4 rectangles with the x axis to be viewed from -1..3 and the y axis to be viewed from -1..6. leftbox(f(x), x = 0..2, 4, view = [-1..3, -1..6]);

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The compute the areas of these rectangles, we use the leftsum command. This command has the syntax leftsum(function, x-axis interval, number of rectangles) where where function is the function to be used, x-axis interval givens the endpoints of the definite integral displayed in the the syntax x = lowerend..upperend, and number of rectangles is the number of rectangles (subintervals) to be used. The next command computes the left endpoint sum for the function f(x) entered above over the interval from x = 0..2 with 4 subintervals (rectangles). Note that we enclose the command in evalf to convert the result to decimal. evalf(leftsum(f(x), x = 0..2, 4));

JCIrKysrXVAhIio=

To draw the rectangles using midpoints, we use the middlebox command. The syntax for this command is almost identical to leftbox. Then next command draws the rectangles using midpoints for the function f(x) given above for the interval x = 0..2 using 4 rectangles. middlebox(f(x), x = 0..2, 4, view = [-1..3, -1..6]);

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Using the middlesum, which has a syntax similar to leftsum, we compute the midpoint sum for 4 subintervals. evalf(middlesum(f(x), x = 0..2, 4));

JCIrKysrRFkhIio=

The next two commands use the rightbox and rightsum commands draw rectangles using right endpoints with 4 rectangles and compute the following right endpoint sum. rightbox(f(x), x = 0..2, 4, view = [-1..3, -1..6]);

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evalf(rightsum(f(x), x = 0..2, 4));

JCIrKysrXWQhIio=

Recall that increasing the number of subintervals (rectangles) will increase the accuracy of the area approximation. The next sequence of commands shows the rectangle graphs and calculations for left endpoint sums, midpoint sums, and right endpoint sums for 30 subintervals (rectangles). leftbox(f(x), x = 0..2, 30, view = [-1..3, -1..6]);

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evalf(leftsum(f(x), x = 0..2, 30));

JCIrOlsiW2AlISIq

middlebox(f(x), x = 0..2, 30, view = [-1..3, -1..6]);

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evalf(middlesum(f(x), x = 0..2, 30));

JCIrRWYjZm0lISIq

rightbox(f(x), x = 0..2, 30, view = [-1..3, -1..6]);

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evalf(rightsum(f(x), x = 0..2, 30));

JCIrI1siWyxbISIq

The next sequence of commands computes the sums for 100 subintervals (rectangles). leftbox(f(x), x = 0..2, 100, view = [-1..3, -1..6]);

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evalf(leftsum(f(x), x = 0..2, 100));

JCIrKyshb2klISIq

middlebox(f(x), x = 0..2, 100, view = [-1..3, -1..6]);

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evalf(middlesum(f(x), x = 0..2, 100));

JCIrKytnbVkhIio=

rightbox(f(x), x = 0..2, 100, view = [-1..3, -1..6]);

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$""!!""7$$"#=!""$"&CJ%!"%7$$"$#=!"#$"&CJ%!"%7$$"$#=!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$#=!"#$""!!""7$$"$#=!"#$"&cQ%!"%7$$"$%=!"#$"&cQ%!"%7$$"$%=!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$%=!"#$""!!""7$$"$%=!"#$"&'fW!"%7$$"$'=!"#$"&'fW!"%7$$"$'=!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$'=!"#$""!!""7$$"$'=!"#$"&W`%!"%7$$"$)=!"#$"&W`%!"%7$$"$)=!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$)=!"#$""!!""7$$"$)=!"#$"$h%!"#7$$"#>!""$"$h%!"#7$$"#>!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"#>!""$""!!""7$$"#>!""$"&ko%!"%7$$"$#>!"#$"&ko%!"%7$$"$#>!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$#>!"#$""!!""7$$"$#>!"#$"&Ow%!"%7$$"$%>!"#$"&Ow%!"%7$$"$%>!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$%>!"#$""!!""7$$"$%>!"#$"&;%[!"%7$$"$'>!"#$"&;%[!"%7$$"$'>!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$'>!"#$""!!""7$$"$'>!"#$"&/#\!"%7$$"$)>!"#$"&/#\!"%7$$"$)>!"#$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"$)>!"#$""!!""7$$"$)>!"#$"#]!""7$$"#?!""$"#]!""7$$"#?!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%%VIEWG6$;$!#5!""$"#I!"";$!#5!""$"#g!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"x6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%%ROOTG6'-%)BOUNDS_XG6#$"$I"!""-%)BOUNDS_YG6#$"$?"!""-%-BOUNDS_WIDTHG6#$"%]Y!""-%.BOUNDS_HEIGHTG6#$"%5Y!""-%)CHILDRENG6"

evalf(rightsum(f(x), x = 0..2, 100));

JCIrKyshb3ElISIq

Without viewing the rectangles, the next sequence of commands compute each type of sum for 1000 rectangles. evalf(leftsum(f(x), x = 0..2, 1000));

JCIrKyFvRW0lISIq

evalf(middlesum(f(x), x = 0..2, 1000));

JCIrK2dtbVkhIio=

evalf(rightsum(f(x), x = 0..2, 1000));

JCIrKyFvMW4lISIq

Finding Exact Areas The exact values for the leftsum, middlesum, and rightsum commands can be found. To demonstrate, we first store the results of the leftsum command in the variable l. Note that number of subintervals is specified as n. l := Limit( leftsum(f(x), x = 0..2, n), n = infinity );

LUkmTGltaXRHSShfc3lzbGliRzYiNiQsJComSSJuR0YlISIiLUkkU3VtR0YkNiQsJiomSSJpR0YlIiIjRikhIiMiIiUiIiJGNC9GMDsiIiEsJkYpRjRGKkY0RjRGMS9GKUkpaW5maW5pdHlHJSpwcm90ZWN0ZWRH

To evaluate the summation and limit, we use the value command. The next command uses value to find the exact result of the previous left endpoint summation for n subintervals as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJ0Y+RitGV0Y+ stored in the variable l. value(l);

IyIjOSIiJA==

The evalf command can be used to find a numerical approximation for the sum. evalf( value(l) );

JCIrbm1tbVkhIio=

The next sequence of commands evaluates 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 using the midpoint sum. m := Limit( middlesum(f(x), x = 0..2, n), n = infinity );

LUkmTGltaXRHSShfc3lzbGliRzYiNiQsJComSSJuR0YlISIiLUkkU3VtR0YkNiQsJiomLCZJImlHRiUiIiIjRjIiIiNGMkY0RikhIiMiIiVGMkYyL0YxOyIiISwmRilGMkYqRjJGMkY0L0YpSSlpbmZpbml0eUclKnByb3RlY3RlZEc=

value(m);

IyIjOSIiJA==

evalf( value(m) );

JCIrbm1tbVkhIio=

The next sequence of commands evaluates 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 using the right endpoint sum. r := Limit( rightsum(f(x), x = 0..2, n), n = infinity);

LUkmTGltaXRHSShfc3lzbGliRzYiNiQsJComSSJuR0YlISIiLUkkU3VtR0YkNiQsJiomSSJpR0YlIiIjRikhIiMiIiUiIiJGNC9GMDtGNEYpRjRGMS9GKUkpaW5maW5pdHlHJSpwcm90ZWN0ZWRH

value(r);

IyIjOSIiJA==

evalf( value(r) );

JCIrbm1tbVkhIio=

In each case, we see that 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.

In this example, we approximate the definite integral 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. The next command enters 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 as a Maple function. g := x -> exp(-x^2);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJCokOSQiIiMhIiJGJUYlRiU=

The next sequence of commands shows the rectangle graphs and calculations for left endpoint sums, midpoint sums, and right endpoint sums for 5 subintervals (rectangles). leftbox(g(x), x = -1..3, 5, view = [-2..4, -1..2]);

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

evalf(leftsum(g(x), x = -1..3, 5));

JCIrWCYpM1M8ISIq

middlebox(g(x), x = -1..3, 5, view = [-2..4, -1..2]);

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evalf(middlesum(g(x), x = -1..3, 5));

JCIrbFlMYDshIio=

rightbox(g(x), x = -1..3, 5, view = [-2..4, -1..2]);

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evalf(middlesum(g(x), x = -1..3, 5));

JCIrbFlMYDshIio=

The next sequence of commands shows the rectangle graphs and calculations for left endpoint sums, midpoint sums, and right endpoint sums for 30 subintervals (rectangles). leftbox(g(x), x = -1..3, 30, view = [-2..4, -1..2]);

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evalf(leftsum(g(x), x = -1..3, 30));

JCIrL29YYzshIio=

middlebox(g(x), x = -1..3, 30, view = [-2..4, -1..2]);

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evalf(middlesum(g(x), x = -1..3, 30));

JCIrI2Z4TmoiISIq

rightbox(g(x), x = -1..3, 30, view = [-2..4, -1..2]);

-%%PLOTG6D-%'CURVESG6&7W7$$!#5!""$"2LU9<T%zyO!#<7$$!1LLLLQ6G"*!#;$"2:FY-dakM%!#<7$$!1nmmT.\p$)!#;$"2%zP91<Xj\!#<7$$!1LLL$))Qj^(!#;$"1uHI\d'Qo&!#;7$$!1LLL$=Kvl'!#;$"1$R02U7'>k!#;7$$!1ommTs!G!e!#;$"1#*3/&yY59(!#;7$$!1MLL3yO5]!#;$"1@s(\>H*zx!#;7$$!21++]nU)*=%!#<$"1TMu$ys**Q)!#;7$$!1MLL3WDTL!#;$"1*)*R6TjO%*)!#;7$$!1,++vvQ&\#!#;$"1N#4^O&H'R*!#;7$$!2immmc4`i"!#<$"1AuinaHR(*!#;7$$!2(*****\(p7U7!#<$"12pZ)>'*o%)*!#;7$$!1KLLLQW*e)!#<$"1ElA]H\E**!#;7$$!0nmmT:dF%!#;$"1[=ge\t")**!#;7$$"2.B*******H,Q!#?$"1N>,b&)******!#;7$$"1&*******RXpV!#<$"1Cnh&3E4)**!#;7$$"1(*******\*3q)!#<$"1rU5r-eC**!#;7$$"2)*******4/vG"!#<$"1ZH])o*fN)*!#;7$$")(=\q"!")$"1UF%p<4Nr*!#;7$$"2lmm;fBIY#!#<$"0FV_2'Q6%*!#:7$$"1LLLLO[kL!#;$"1CP;#zU(H*)!#;7$$"1KLLL&Q"GT!#;$"1nT9V`9L%)!#;7$$"1*****\s]k,&!#;$"1Eq*Q[&=vx!#;7$$"1JLLLvv-e!#;$"1;*y8)z3Tr!#;7$$"1,++D2Ylm!#;$"1uK7U[$GT'!#;7$$"+v"ep[(!#5$"1_!4(ef(*3d!#;7$$"1MLL$e/TM)!#;$"2'G#p'Rdb%)\!#<7$$"1LLLeDBJ"*!#;$"2myY![,)RM%!#<7$$"1mmm;kD!)**!#;$"2%QT()\_L$p$!#<7$$"2jmm"f`@'3"!#;$"2ad6#3f?tI!#<7$$"2)****\nZ)H;"!#;$"1yqq()G%ee#!#;7$$"2lmm;$y*eC"!#;$"2s^$)3P&o<@!#<7$$"2*******R^bJ8!#;$"2:vN?<h")p"!#<7$$"2'*****\5a`T"!#;$"2nz1YF#**[8!#<7$$"2(****\7RV'\"!#;$"2s+?b6:`1"!#<7$$"2'*****\@fke"!#;$"1M<`"[A82)!#<7$$"1LLL`4Nn;!#:$"1-=$3Z$[.i!#<7$$"*:?Pv"!")$"/4n(G&\;Y!#:7$$"2lmm"zM)>$=!#;$"2whHSyxo[$!#=7$$"*(fa<>!")$"2$*Q<=*)>*HD!#=7$$"1LL$eg`!)*>!#:$"1m:fJB(e%=!#<7$$"2)****\#G2A3#!#;$"2#)Q6RL=%48!#=7$$"1LLL$)G[k@!#:$"1-CylC'GB*!#=7$$",D"yh]A!#5$"1$)fmf)R@J'!#=7$$"2ommm)fdLB!#;$"2&34[E)\`J%!#>7$$"2mmm;q7%=C!#;$"1y'o>!ye$)G!#=7$$"2LLLe#pa-D!#;$"2,SFR-9g!>!#>7$$"*ad)zD!")$"2mOJ%*G7nG"!#>7$$"2LLL$GUYoE!#;$"1H;87t*>3)!#>7$$"2lmmm5:xu#!#;$"1&ym@&eUh_!#>7$$",D28A$G!#5$"2Z]`ja)\$G$!#?7$$",vS)38H!#5$"2OxdX8'>j?!#?7$$"#I!""$"2czm3/)4M7!#?-%&COLORG6&%$RGBG$"#5!""$""!!""$""!!""-%*THICKNESSG6#""#-%&STYLEG6#%%LINEG-%)POLYGONSG6$7&7$$!#5!""$""!!""7$$!#5!""$"+D*>%=Z!#57$$!+nmmm')!#5$"+D*>%=Z!#57$$!+nmmm')!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$!+nmmm')!#5$""!!""7$$!+nmmm')!#5$"+%)oWSe!#57$$!+LLLLt!#5$"+%)oWSe!#57$$!+LLLLt!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$!+LLLLt!#5$""!!""7$$!+LLLLt!#5$"+hKwwp!#57$$!"'!""$"+hKwwp!#57$$!"'!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$!"'!""$""!!""7$$!"'!""$"*cTI/)!"*7$$!+nmmmY!#5$"*cTI/)!"*7$$!+nmmmY!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$!+nmmmY!#5$""!!""7$$!+nmmmY!#5$"+oJR[*)!#57$$!+LLLLL!#5$"+oJR[*)!#57$$!+LLLLL!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$!+LLLLL!#5$""!!""7$$!+LLLLL!#5$"+#R%*yg*!#57$$!"#!""$"+#R%*yg*!#57$$!"#!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$!"#!""$""!!""7$$!"#!""$"+vTlb**!#57$$!+nmmmm!#6$"+vTlb**!#57$$!+nmmmm!#6$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$!+nmmmm!#6$""!!""7$$!+nmmmm!#6$"+vTlb**!#57$$"+nmmmm!#6$"+vTlb**!#57$$"+nmmmm!#6$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmmm!#6$""!!""7$$"+nmmmm!#6$"+#R%*yg*!#57$$""#!""$"+#R%*yg*!#57$$""#!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$""#!""$""!!""7$$""#!""$"+oJR[*)!#57$$"+LLLLL!#5$"+oJR[*)!#57$$"+LLLLL!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+LLLLL!#5$""!!""7$$"+LLLLL!#5$"*cTI/)!"*7$$"+nmmmY!#5$"*cTI/)!"*7$$"+nmmmY!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmmY!#5$""!!""7$$"+nmmmY!#5$"+hKwwp!#57$$""'!""$"+hKwwp!#57$$""'!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$""'!""$""!!""7$$""'!""$"+%)oWSe!#57$$"+LLLLt!#5$"+%)oWSe!#57$$"+LLLLt!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+LLLLt!#5$""!!""7$$"+LLLLt!#5$"+D*>%=Z!#57$$"+nmmm')!#5$"+D*>%=Z!#57$$"+nmmm')!#5$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmm')!#5$""!!""7$$"+nmmm')!#5$"+7WzyO!#57$$"#5!""$"+7WzyO!#57$$"#5!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"#5!""$""!!""7$$"#5!""$"+7K/oF!#57$$"+LLLL6!"*$"+7K/oF!#57$$"+LLLL6!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+LLLL6!"*$""!!""7$$"+LLLL6!"*$"+8>,5?!#57$$"+nmmm7!"*$"+8>,5?!#57$$"+nmmm7!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmm7!"*$""!!""7$$"+nmmm7!"*$"+4Ue39!#57$$"#9!""$"+4Ue39!#57$$"#9!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"#9!""$""!!""7$$"#9!""$"+MbKE&*!#67$$"+LLLL:!"*$"+MbKE&*!#67$$"+LLLL:!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+LLLL:!"*$""!!""7$$"+LLLL:!"*$"+,Cl<i!#67$$"+nmmm;!"*$"+,Cl<i!#67$$"+nmmm;!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmm;!"*$""!!""7$$"+nmmm;!"*$"*^*Q;R!#57$$"#=!""$"*^*Q;R!#57$$"#=!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"#=!""$""!!""7$$"#=!""$"+s[p!Q#!#67$$"+LLLL>!"*$"+s[p!Q#!#67$$"+LLLL>!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+LLLL>!"*$""!!""7$$"+LLLL>!"*$"+Zci'R"!#67$$"+nmmm?!"*$"+Zci'R"!#67$$"+nmmm?!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmm?!"*$""!!""7$$"+nmmm?!"*$"+_S02z!#77$$"#A!""$"+_S02z!#77$$"#A!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"#A!""$""!!""7$$"#A!""$"+w%R-K%!#77$$"+LLLLB!"*$"+w%R-K%!#77$$"+LLLLB!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+LLLLB!"*$""!!""7$$"+LLLLB!"*$"+Y&H!yA!#77$$"+nmmmC!"*$"+Y&H!yA!#77$$"+nmmmC!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmmC!"*$""!!""7$$"+nmmmC!"*$"+u"H#f6!#77$$"#E!""$"+u"H#f6!#77$$"#E!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"#E!""$""!!""7$$"#E!""$"*&R&Hp&!#77$$"+LLLLF!"*$"*&R&Hp&!#77$$"+LLLLF!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+LLLLF!"*$""!!""7$$"+LLLLF!"*$"+e)R")p#!#87$$"+nmmmG!"*$"+e)R")p#!#87$$"+nmmmG!"*$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%)POLYGONSG6$7&7$$"+nmmmG!"*$""!!""7$$"+nmmmG!"*$"+T!)4M7!#87$$"#I!""$"+T!)4M7!#87$$"#I!""$""!!""-%&COLORG6&%$RGBG$"1rCy?!3'>q!#;$"1"=t:!zg>!*!#;$"1rCy?!3'>q!#;-%%VIEWG6$;$!#?!""$"#S!"";$!#5!""$"#?!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"x6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%%ROOTG6'-%)BOUNDS_XG6#$"$I"!""-%)BOUNDS_YG6#$"$?"!""-%-BOUNDS_WIDTHG6#$"%]Y!""-%.BOUNDS_HEIGHTG6#$"%5Y!""-%)CHILDRENG6"

evalf(middlesum(g(x), x = -1..3, 30));

JCIrI2Z4TmoiISIq

The next sequence of commands shows the calculations for left endpoint sums, midpoint sums, and right endpoint sums for 100 subintervals (rectangles). evalf(leftsum(g(x), x = -1..3, 100));

JCIrKFIpR1M7ISIq

evalf(middlesum(g(x), x = -1..3, 100));

JCIrJ2UhM0w7ISIq

evalf(rightsum(g(x), x = -1..3, 100));

JCIrYiJ5YmkiISIq

Finding Exact Areas The exact values for the leftsum, middlesum, and rightsum commands can be found. To demonstrate, we first store the results of the leftsum command in the variable l. Note that number of subintervals is specified as n. l := Limit(leftsum(g(x), x = -1..3, n), n = infinity);

LUkmTGltaXRHSShfc3lzbGliRzYiNiQsJComSSJuR0YlISIiLUkkU3VtR0YkNiQtSSRleHBHNiQlKnByb3RlY3RlZEdGJDYjLCQqJCwmRioiIiIqJkkiaUdGJUY2RilGKiIiJSIiI0YqL0Y4OyIiISwmRilGNkYqRjZGNkY5L0YpSSlpbmZpbml0eUdGMQ==

To evaluate the summation and limit, we use the value command. The next command uses value to find the result of the previous left endpoint summation for n subintervals as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJ0Y+RitGV0Y+ stored in the variable l. value(l);

LCYqJkkjUGlHJSpwcm90ZWN0ZWRHIyIiIiIiIy1JJGVyZkc2JEYlSShfc3lzbGliRzYiNiNGJ0YnRiYqJkYkRiYtRio2IyIiJEYnRiY=

The previous output is Maple's way of indicating that an exact solution does not exist. However, using the evalf command, we can find an approximate solution to 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 evalf( value(l) );

JCIrI1tKSWoiISIq

The next sequence of commands evaluates 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 using the midpoint sum. m := Limit( middlesum(g(x), x = -1..3, n), n = infinity);

LUkmTGltaXRHSShfc3lzbGliRzYiNiQsJComSSJuR0YlISIiLUkkU3VtR0YkNiQtSSRleHBHNiQlKnByb3RlY3RlZEdGJDYjLCQqJCwmRioiIiIqJiwmSSJpR0YlRjYjRjYiIiNGNkY2RilGKiIiJUY7RiovRjk7IiIhLCZGKUY2RipGNkY2RjwvRilJKWluZmluaXR5R0Yx

value(m);

LCYqJkkjUGlHJSpwcm90ZWN0ZWRHIyIiIiIiIy1JJGVyZkc2JEYlSShfc3lzbGliRzYiNiNGJ0YnRiYqJkYkRiYtRio2IyIiJEYnRiY=

evalf( value(m) );

JCIrI1tKSWoiISIq

The next sequence of commands approximates 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 using the right endpoint sum. r := Limit(rightsum(g(x), x = -1..3, n), n = infinity);

LUkmTGltaXRHSShfc3lzbGliRzYiNiQsJComSSJuR0YlISIiLUkkU3VtR0YkNiQtSSRleHBHNiQlKnByb3RlY3RlZEdGJDYjLCQqJCwmRioiIiIqJkkiaUdGJUY2RilGKiIiJSIiI0YqL0Y4O0Y2RilGNkY5L0YpSSlpbmZpbml0eUdGMQ==

value(r);

LCYqJkkjUGlHJSpwcm90ZWN0ZWRHIyIiIiIiIy1JJGVyZkc2JEYlSShfc3lzbGliRzYiNiNGJ0YnRiYqJkYkRiYtRio2IyIiJEYnRiY=

evalf( value(r) );

JCIrI1tKSWoiISIq

Hence, 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 is approximately equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEsMS42MzMwMzE0ODJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw==.

1. Use left endpoint point sums, midpoint sums, and right endpoint sums to approximate the following definite integrals using n = 5 and n = 75 subintervals (rectangles). Use the leftbox, middlebox, and rightbox commands to clearly draw the rectangles under the curve and the leftsum, middlesum, and rightsum commands to find the left endpoint, right endpoint, and middlepoint sums. Then, in conjuction with leftsum, middlesum, and rightsum, and the limit command, let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg== and find the exact value or a very good approximate value for the definite integral using left endpoints, midpoints, and right endpoints. a. 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 b. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LlErJkludGVncmFsO0YnLyUrZm9yZWdyb3VuZEdRLlsxNDQsMTQ0LDE0NF1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi1GIzYoLUYvNi1RKiZ1bWludXMwO0YnRjUvRjlRJmZhbHNlRicvRjxGUy9GPkZTL0ZBRlMvRkNGUy9GRUZTL0ZHRlMvRklRLDAuMjIyMjIyMmVtRicvRkxGZW4tSSNtbkdGJDYkUSI0RidGNS8lJ2l0YWxpY0dGPy9GM1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y/L0Y2USdpdGFsaWNGJy1GIzYlRmduRltvRmFvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0Znby1GLzYtUSJ+RidGNUZSRlRGVUZWRldGWEZZRkhGSy1JJW1zdXBHRiQ2JS1JI21pR0YkNiVRInhGJ0Zbb0Zhby1GIzYlLUZobjYkUSIyRidGNUZbb0Zhb0Zlby1GXnA2JS1GYXA2JVEiZUYnRltvRmFvLUYjNiZGT0ZdcEZbb0Zhb0Zlb0Zqby1GLzYuUTAmRGlmZmVyZW50aWFsRDtGJ0YyRjVGOEY7L0Y+RjpGQC9GQ0Y6RkRGRkZIRkstRmFwNidGY3BGW29GXW9GX29GYW8tRi82LkZccEYyRjVGUkZURlVGVkZXRlhGWUZIRktGNQ==