<Text-field style="Heading 1" layout="Heading 1">Review of Maple Plotting</Text-field>

Lab1: Integrals - Boxes and Boxes

Jeremy Johnson

This lab explores the definition of the definite integral. We begin by approximating the area under a curve by a sum of the areas of rectangles covering the curve. The curve is given by a function y=f(x) defined on an interval [a,b], with a < b, and where [a,b] denotes the set of numbers between a and b. The interval [a,b] is partitioned into N subintervals of width (b-a)/N = h: [a,a+h],[a+h,a+2*h],...,[a+(N-1)*h,a+N*h=b]. The base of the ith rectangle is the subinterval [a+(i-1)*h,a+i*h] and the height of the rectangle is equal to f(a+i*h). As the number of rectangles increases the sum of the areas of the rectangles approaches the area under the curve. The area under the curve defined by the function f(x) over the interval [a,b] can be evaluated in Maple using the definite integration command: int(f(x),x=a..b). The following plot shows this procedure for the curve y = sin(x) on the intervap [0..2*Pi] using 20 rectangles.

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In this lab you will estimate the area under a curve by calculating the sum of the areas of the rectangles as more and more rectangles are used. You will see the approximate area approaching the area under the curve and will see that in the limit as the width of the rectangles approaches 0, a formula for the are under the curve is obtained. These examples hint at the fundamental theorem of calculus which shows how to evaluate definite integrals (area under a curve) with a process called anti-differentiation.

In order to do the lab, you will need to review Maple's plotting capabilities, the use of the commands sum, seq, limit, and evalf, and Maple procedures and functions. These are reviewed below.

<Text-field style="Heading 1" layout="Heading 1">Review of Maple Plotting</Text-field>

This section reviews some of Maple's plotting tools. In addition to the basic plot routine, we review how to assign plots to variables and display multiple plots using the display command from the plots package. Recall that the with command enables use to access commands from a package without having to use the package name.

We also introduce some lower level commands in the plottools package which can be used to construct plots from components. In particular, we use the rectangle command to build a plot with a collection of rectangles.

All Maple plots, no matter how they are produced, are stored in a data structore called a plot structure. See help on plot[structure] to learn more. Below I show how to construct a plot from a sequence of rectangles. This example leads into what we will be covering in this lab; namely approximating the area under a curve with rectangles.

with(plots);

Warning, the name changecoords has been redefined

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FaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q*rootlocusF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q,semilogplotF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q+setoptionsF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q-setoptions3dF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q+spacecurveF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q1sparsematrixplotF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q+sphereplotF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q)surfdataF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q)textplotF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q+textplot3dF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfqFhq-Fgn69Q)tubeplotF(FjnF]oF_oFaoFcoFeoFgoFioF[pF]pF_pFapFcpFepFgpFipF[qF]qF_qFaqFcqFfq-F-63Q"]F(/F1Q(postfixF(F3F6F9/F=Q2verythinmathspaceF(F>F@FBFEFHFJFLFNFQFTFW7#7gnI,InteractiveGF(I(animateGF(I*animate3dGF(I-animatecurveGF(I&arrowGF(I-changecoordsGF*I,complexplotGF(I.complexplot3dGF(I*conformalGF(I,conformal3dGF(I,contourplotGF(I.contourplot3dGF(I*coordplotGF(I,coordplot3dGF(I-cylinderplotGF(I,densityplotGF(I(displayGF(I*display3dGF(I*fieldplotGF(I,fieldplot3dGF(I)gradplotGF(I+gradplot3dGF(I,graphplot3dGF(I-implicitplotGF(I/implicitplot3dGF(I(inequalGF(I,interactiveGF(I2interactiveparamsGF(I-listcontplotGF(I/listcontplot3dGF(I0listdensityplotGF(I)listplotGF(I+listplot3dGF(I+loglogplotGF(I(logplotGF(I+matrixplotGF(I)multipleGF(I(odeplotGF(I'paretoGF(I,plotcompareGF(I*pointplotGF(I,pointplot3dGF(I*polarplotGF(I,polygonplotGF(I.polygonplot3dGF(I4polyhedra_supportedGF(I.polyhedraplotGF(I'replotGF(I*rootlocusGF(I,semilogplotGF(I+setoptionsGF(I-setoptions3dGF(I+spacecurveGF(I1sparsematrixplotGF(I+sphereplotGF(I)surfdataGF(I)textplotGF(I+textplot3dGF(I)tubeplotGF(

with(plottools);

Warning, the assigned name arrow now has a global binding

plot(x^2,x=0..1);

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

C := plot(x^2,x=0..1):

B1 := rectangle([0,0],[0.5,0.5^2],transparency=1);

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display(B1);

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B2 := rectangle([0.5,0],[1,1],transparency=1):

display({B1,B2,C});

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N := 4; h := 1/N;

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

polys := seq(rectangle([(i-1)*h,0],[i*h,(i*h)^2],transparency=1),i=1..N);

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

boxes := PLOT(polys);

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

display(boxes,C);

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<Text-field style="Heading 1" layout="Heading 1">Review of limits, sequences, sums, and evalf</Text-field>

In order to do this, lab you must be comfortable using the seq, sum, limit, and evalf commands. This section reviews these commands in a context that is relevent for this lab.

Recall that Maple can determine a formula for a sum in terms of parameters defining the range of the summation indices. This is called symbolic summation.

sum(i,i=1..n);

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S := simplify(%);

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The sum command can also be used when the range parameters are known. However, Maple first finds a formula for the sum as above and then plugs in the particular value for the range parameter. The command add does not use "symbolic summation", but rather simply adds up the elements in a fixed range. This will work whether or not a formula can be obtained by sum and is usually much faster than using sum.

sum(i,i=1..10);

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add(i,i=1..10);

From above summation formula, we know that the sum of the first n integers is approximately 1/2*n^2 for large n. Let's investigate this using a sequence of ratios. Sometimes it is useful to approximate the ratios using floating point arithmetic rather than fractions. This can be done through the use of evalf. Maple's limit command can be used to determine the limiting ratio as n goes to infinity.

seq(S/n^2,n=1..20);

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

seq(evalf(S/n^2),n=1..20);

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

limit(S/n^2,n=infinity);

NiQtSSZtZnJhY0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGKDYqLUkjbW5HRiU2OVEiMUYoLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRigvJSVzaXplR1EjMTJGKC8lJWJvbGRHUSZmYWxzZUYoLyUnaXRhbGljR0Y4LyUqdW5kZXJsaW5lR0Y4LyUqc3Vic2NyaXB0R0Y4LyUsc3VwZXJzY3JpcHRHRjgvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRigvJStiYWNrZ3JvdW5kR1EoWzAsMCwwXUYoLyUnb3BhcXVlR0Y4LyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R1EldHJ1ZUYoLyUpY29tcG9zZWRHRjgvJSpjb252ZXJ0ZWRHRjgvJStpbXNlbGVjdGVkR0Y4LyUscGxhY2Vob2xkZXJHRjgvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGKC8lKm1hdGhjb2xvckdGQy8lL21hdGhiYWNrZ3JvdW5kR0ZGLyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRigvJSltYXRoc2l6ZUdGNS1GLTY5USIyRihGMEYzRjZGOUY7Rj1GP0ZBRkRGR0ZJRktGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8vJS5saW5ldGhpY2tuZXNzR1EiMUYoLyUrZGVub21hbGlnbkdRJ2NlbnRlckYoLyUpbnVtYWxpZ25HRmZvLyUpYmV2ZWxsZWRHRjgvJStmb3JlZ3JvdW5kR0ZDLyUrYmFja2dyb3VuZEdGRjcjIyIiIiIiIw==

S/n^2;

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

map((x)->x/n^2,S);

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

limit(1/2+1/n,n=infinity);

limit(1/2+e,e=0);

<Text-field style="Heading 1" layout="Heading 1">Review of Maple Procedures</Text-field>

In this lab, you will approximate the area under a curve. This will first be done for two specific curves and then you are to write a Maple procedure which can be used for arbitrary curves. In this section, we review how to write a Maple procedure (the example is similar to one that you will have to write).

The following procedure, plotboxes, plots the rectangles discussed in the introduction used to estimate the area under the curve y = x^2 on the interval [0,1]. The parameter N indicates the number of rectangles that are used. The procedure creates a sequence of rectangles and then adds them and the curve y = x^2 into a plot data structure which is returned. The plot data structure will be displayed by Maple if the call to the procedure ends in a semicolon.

with(plots);

plotboxes := proc(N)

local polys, h, x, C; h := 1/N;

polys := seq(plottools[rectangle]([(i-1)*h,0],[i*h,(i*h)^2],transparency=1),i=1..N);

C := plot(x^2,x=0..1);

return `PLOT`(polys,op(C));

end;

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USpwbG90Ym94ZXNGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYoLyUrYmFja2dyb3VuZEdRKFswLDAsMF1GKC8lJ29wYXF1ZUdGOC8lK2V4ZWN1dGFibGVHRjgvJSlyZWFkb25seUdGOy8lKWNvbXBvc2VkR0Y4LyUqY29udmVydGVkR0Y4LyUraW1zZWxlY3RlZEdGOC8lLHBsYWNlaG9sZGVyR0Y4LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRy8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjUtSSNtb0dGJTYzUSM6PUYoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRigvJSdyc3BhY2VHRltwLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGWC8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRy1GJDYpLUZfbzYzUSVwcm9jRigvRmNvUSFGKEZlb0Znby9Gam9RJDBlbUYoL0ZdcEZeckZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcS1GJDYlLUZfbzYzUSIoRigvRmNvUSdwcmVmaXhGKC9GZm9GO0Znby9Gam9RLnRoaW5tYXRoc3BhY2VGKC9GXXBGaXIvRl9wRjtGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2Iy1GLTY5USJORihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tRl9vNjNRIilGKC9GY29RKHBvc3RmaXhGKEZnckZnb0Zoci9GXXBRMnZlcnl0aGlubWF0aHNwYWNlRihGW3NGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtSSdtc3BhY2VHRiU2Ji8lJ2hlaWdodEdRJzAuMH5leEYoLyUmd2lkdGhHUScwLjN+ZW1GKC8lJmRlcHRoR0ZddC8lKmxpbmVicmVha0dRMWZpcnN0cHJvY25ld2xpbmVGKC1GJDYmLUZfbzYzUSZsb2NhbEYoRltyRmVvRmdvRl1yL0ZdcFEwbWVkaXVtbWF0aHNwYWNlRihGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2KS1GLTY5USZwb2x5c0YoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZfbzYzUSIsRihGYm9GZW8vRmhvRjtGXXIvRl1wUTN2ZXJ5dGhpY2ttYXRoc3BhY2VGKEZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcS1GLTY5USJoRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GYnUtRi02OVEieEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRmJ1LUYtNjlRIkNGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1GX282M1EiO0YoRmJvRmVvRmV1Rl1yRlxwRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxRmhzLUYkNiYtRiQ2JS1GJDYlRmh1Rl5vLUYkNiUtSSNtbkdGJTY5RmdwRjBGM0Y2L0Y6RjhGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduL0ZqblEnbm9ybWFsRihGXG8tRl9vNjNRIi9GKEZib0Zlb0Znb0ZockZqckZbc0ZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcUZcc0ZhdkZocy1GJDYlLUYkNiVGX3VGXm8tRiQ2JS1GLTY5USRzZXFGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1GX282M1EwJkFwcGx5RnVuY3Rpb247RihGYm9GZW9GZ29GXXJGX3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2JUZici1GJDYlLUYkNiUtRiQ2Ji1GLTY5USpwbG90dG9vbHNGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1GX282M1EiW0YoRmVyRmdyRmdvRmhyRmpyRltzRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYkNiMtRi02OVEqcmVjdGFuZ2xlRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tRl9vNjNRIl1GKEZkc0ZnckZnb0ZockZmc0Zbc0ZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcUZeeC1GJDYlRmJyLUYkNictRiQ2JUZceS1GJDYlLUYkNiUtRiQ2JUZici1GJDYlLUYtNjlRImlGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1GX282M1EqfiZtaW51czt+RihGW3JGZW9GZ29GXXJGX3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHFGXHdGYXMtRl9vNjNRIipGKEZib0Zlb0Znb0ZockZqckZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcUZodUZidS1GXXc2OVEiMEYoRjBGM0Y2Rl93RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZgd0Zcb0ZkeUZidS1GJDYlRlx5LUYkNiUtRiQ2JUZlekZbW2xGaHVGYnUtRiQ2JS1GJDYlRmV6LUZfbzYzUSJeRihGYm9GZW9GZ28vRmpvRmdzRmZzRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUZddzY5USIyRihGMEYzRjZGX3dGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmB3RlxvRltbbC1GJDYlRmh1RltcbEZfXGxGZHlGYnUtRiQ2JS1GLTY5US10cmFuc3BhcmVuY3lGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1GX282M1EiPUYoRmJvRmVvRmdvRmlvRlxwRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxRlx3RmFzRmJ1LUYkNiVGZXpGaVxsLUYkNiVGXHctRl9vNjNRIy4uRihGZHNGZW9GZ28vRmpvRlx1Rl9yRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxRl5zRmFzRmF2RmhzLUYkNiUtRiQ2JUZedkZeby1GJDYlLUYtNjlRJXBsb3RGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcb0ZeeC1GJDYlRmJyLUYkNiUtRiQ2Iy1GJDYlRlt2RltcbEZfXGxGYnUtRiQ2JUZbdkZpXGwtRiQ2JUZeW2xGYF1sRlx3RmFzRmF2RmhzLUYkNiQtRl9vNjNRJ3JldHVybkYoRltyRmVvRmdvRl1yRlt1Rl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYkNiUtRi02OVElUExPVEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRl54LUYkNiVGYnItRiQ2JUZfdUZidS1GJDYlLUYtNjlRI29wRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GXngtRiQ2JUZici1GJDYjRl52RmFzRmFzLUZpczYmRlt0L0ZfdFEnMC4wfmVtRihGYXQvRmR0UTZkZWNyZWFzZWluZGVudG5ld2xpbmVGKC1GX282M1EpZW5kfnByb2NGKEZbckZlb0Znb0ZjXWxGX3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHE3Iy1fRilJLG1wcmludHNsYXNoR0YoNiQ3Iz5JKnBsb3Rib3hlc0dGKGYqNiNJIk5HRig2JkkmcG9seXNHRihJImhHRihJInhHRihJIkNHRihGKEYoQyY+RmZhbCokRmNhbCEiIj5GZWFsLUkkc2VxRyUqcHJvdGVjdGVkRzYkLSZJKnBsb3R0b29sc0dGKDYjSSpyZWN0YW5nbGVHRig2JTckKiYsJkkiaUdGKCIiIkZcY2xGXGJsRlxjbEZmYWxGXGNsIiIhNyQq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plotboxes(4);

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plotboxes(8);

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<Text-field style="Heading 1" layout="Heading 1">Problem 0 (Attendance - 2 points)</Text-field>

Hopefully you are here.

<Text-field style="Heading 1" layout="Heading 1">Problem 1 (Approximating the Area Under a Curve - 2 points)</Text-field>

In this problem, you estimate the area under the curves, y = x, x=0..1 and y=x^2, x=0..1 using a sequence of boxes. Given an interval [a,b] and a continuous function f(x) defined on the interval [a,b], the boxes are obtained by partitioning the interval into N subintervals and forming the boxes whose base is the subinterval and whose height is the the function f(x) evaluated at the right endpoint of the subinterval.

For example, given f(x) = x and a=0 and b=1 and N = 4, the subintervals are [0,1/4], [1/4,1/2], [1/2,3/4], [3/4,1] and the boxes are depicted in the following picture.

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Compute the area of the four boxes and compare to the area under the curve. The area under the curve can be obtained with the Maple command int(f(x),x=a..b), this command computes the definite integral of f(x) on the interval [a,b], which is equal to the area under the curve defined by f(x) on the interval [a,b].

Do the computation again using N = 8 (see picture below). Note you should use the maple sum command to compute the sum of the areas of the boxes. What is the area of the i-th box (i=1..N) as a function of i and N? Use this formula in your sum.

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Compute a sequence of areas (computed as a sum of the areas of boxes whose base has width 1/N) as N goes from 1 to 50. Use evalf to get a numerical estimate. Does the sum seem to be approaching the area under the curve?

Do the same for the curve y = x^2.

<Text-field style="Heading 1" layout="Heading 1">Problem 2 (Limiting Boxes and Integration - 2 points)</Text-field>

In this problem you will use maple's limit command to compute the area of the boxes in problem 1 as N goes to infinity. The limiting area should equal the area under the curve (i.e. as the width of the boxes get smaller and smaller, the sum of their areas should get closer and closer to the area under the curve and in the limit should be equal to the area under the curve.

Compute the sum of the area of the boxes (for the curves y=x and y=x^2 on the interval [0,1]) as function of, N, the number of boxes used in the approximation. Then compute the limit of the expression obtained for the sum as N goes to infinity. Compare the result you get to the area under the curve obtained with the int command.

<Text-field style="Heading 1" layout="Heading 1">Problem 3 (Procedures to Plot and Approximate the Area under a Curve using Boxes - 4 points)</Text-field>

In this problem you will generalize what was done in the plotting section and what you did in problem 1. In particular, you will write Maple procedures Approx(f,a,b,N) and PlotBoxes(f,a,b,N), where f is a Maple function, a<b are the endpoints of an interval and N, a positive integer, is the number of subintervals used in the approximation of the area of the curve y = f(x), x=a..b. The input function f should be a Maple function, which for simple functions you can define using the "arrow" notation.

Test your functions Approx and PlotBoxes using the following two functions and the interval [0,1]. You should be able to reproduce your results from Problem 1. Also try the procedure when the interval is [1,2]. Do the answers you get make sense? Check your results using int.

f1 := (x) -> x;

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f2 := (x)->x^2;

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f1(2); f2(2);

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

Fill in the code for Approx and test with f1 and f2 above.

Approx := proc(f,a,b,N)

end;

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