Flick, T. Michael

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I !Ii ,' :III. 'I :1 ,, ! 11 II If Iii' to help the student identify the key mathematical tasks that will be on the test. We need to help the student to develop a plan of attack when taking the test, such as to first work all the prob-lems which seem easiest. Finally, we need to help the student develop an image of successfully coping with the mathematics test. Simply put, the student should verbalize a statement such as, "I can perform well on this test." The stage of confronting the mathematics test would involve these activities by the student: 1. answering one question at a time, not being overwhelmed by the entire test; 2. solving first the questions which are easiest; 3. and taking a deep, slow breath in order to pause for a moment. The stage of coping with the feeling of being overwhelmed involves these activities: 1) remembering the test will be over before too long; 2) pausing for a moment before a difficult ques-tion; 3) labeling fear from Oto 10 and watching it change. Here, the emphasis is on monitoring the flood of anxiety during the test. Reinforcing statements are vital. After the test, the stu-dent should be praised for having taken it (and surviving it!) • The student should verbalize a statement such as, "I am really pleased with the progress I am making." In short, these coping skills need to be practiced and re-fined. Many of the ideas seem to relate to the student developing a sense of confidence with mathematics, a positive feeling which can be extended to the studying of other mathematical concepts. In summary, this article has tried to develop some under-standing of the term "mathematical anxiety." An examination of some of the characteristics of mathematics anxiety has been ex-plored. Also, suggestions for modifying teacher behaviors and teaching strategies have been developed. Finally, strategies for coping with mathematics tests have been developed. Hopefully, these suggestions may help some students to feel more emotionally secure with mathematics. 24 References Meichenbaum, D. A self-instructional approaeh to stress manage-ment: A proposal for stress innoculation training. In I.G. Sarason and C.D. Speilberger (Eds.) Stress and Anxiety - Vol-ume I. New York: John Wiley, 1975. Sarason, S.B. Anxiety in elementary school children. New York: John Wiley & Sons, 1960. SINGLE AND DOUBLE INTEGRALS FOR AREA IN ADVANCED PLACEMENT CALCULUS T. Michael Flick Purcell Marian High School Cincinnati, Ohio One of the most fascinating and useful applications of ele-mentary integral calculus is its application to area. It is at the time of applying integration to area that calculus students begin to see the power of integration. It is the area application that first destroys the myth that integration is simply antidif-ferentiation or reversing the derivative process. Indeed, the geometric interpretation of the integral is light years from the geometric interpretation of the derivative. This fact should be strongly emphasized in the calculus classroom. A firm understanding of just how the area between a curve f(x) and an axis is computed can best be demonstrated using the Riemann integration technique of inscribing 'n' rectangles having equal length bases, finding the sum of their areas, and then letting the number of rectangles approach infinity as a limit. In this way exact areas can be computed and confidence in the funda-mental theorem of integral calculus, b lim L f(x) AX = Ax-+0 a 25 b 5f(x) dx a --·~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...,,, is developed. This intuitive approach is especially needed in Advanced Placement High School calculus before a more rigorous proof of the fundamental theorem is attempted. The student needs to have a feeling for the theorem before formal proof. In this way, a better understanding of the topic is developed. It is easy to build a sound foundation in the student's understanding of this concept. In general, the approach is to take a given function, y = f(x), and inscribe n-rectangles between it and the x-axis (see figure 1). ~.! '1 (a+2t.x,f(a+2Ax) a+Ax,f(a+t.x)) l,. . \t (alf(a))~ , . , 1 = f(x) I A:ic 1 ,t.x, 1- I ~x a b Then that b-a t,x is n where n is~ the total number of rectangles. Note t,x remains constant and is the base of all n of the rectan-b-a gles. Further, emphasis is needed that Ax. n is a function of n and that as n approaches infinity, Ax approaches zero, Con-stants a and bare fixed and not equal. The approximate area, Aa, between f(x) and the x-axis is given by the sum of the areas of then-rectangles drawn under the curve. Aa = f(a) Ax + f(a+ t,x) Ax + f(a+2 Ax) Ax+ ••• + f(a+(n-1) Ax) Ax n-1 n ~ ~ f(a+k•AX) t,x = L f(a+(k-l)•Ax) Ax. 1c=o k=l Of course, as the number of rectangles is increased between a and b, the value of Ax gets smaller, and the value of Aa becomes a better approximation to the exact area 26 A exact· In other words, n-1 n A • lim L f(a+k•t.x) ,h"' lim .L f(a+(k-1) Ax) Ax. exact n-,.o0 k=O n~• k=l However, it is by working specific examples that student confidence in the fundamental theorem can be established. Specif-ic examples will require a knowledge of sigma notation properties (see table 1). The properties are usually proven using mathemati-cal induction in a precalculus course and can be repeated for emphasis in a lecture on this material. TABLE 1 PROPERTIES OF THE SIGMA NOTATION n }: c = en k=l n ~ k = n(n + 1) k=l 2 n ~ k2= n(n + 1)(2n + 12 6 n k3=M2 ~ k=l . 2 As a specific example, consider the area between the graph of f(x) = x2 + 1 and the x-axis on the interval l .S, x .S, 4 (see fig-ure 2). y FIGURE ~ --.. r'"ll C: r'"ll c:: r'"\l.::: + .-, (\J ... er\ + • •• t::i.x r'"\lC 4 .... I ~ .... -27 2 ~f(x) = X + 1 t::i.x = b-a n = 4-1 = 2 n n ... -·~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ii! Ii 11 i\l\.'. :11 ., t is developed. This intuitive approach is especially needed in Advanced Placement High School calculus before a more rigorous proof of the fundamental theorem is at tempted. The student needs to have a feeling for the theorem before formal proof. In this way, a better understanding of the topic is developed. It is easy to build a sound foundation in the student's understanding of this concept. In general, the approach is to take a given function, y = f(x), and inscribe n-rectangles between it and the x-axis (see figure 1). ~.! 1 (a+2Ax,f(a+2Ax) a+Ax,f(a+Ax}) l,. . \t (alf(a})~ , . , 1 = f(x) I Ax 1 ,Ax1 1- I ~z a b Then that b-a Ax is n where n is~ the total number of rectangles. Note Ax remains constant and is the base of all n of the rectan-b-a gles. Further, emphasis is needed that Ax• n is a function of n and that as n approaches infinity, Ax approaches zero. Con-stants a and bare fixed and not equal. The approximate area, Aa, between f(x) and the x-axis is given by the sum of the areas of then-rectangles drawn under the curve. Aa = f(a) t,x + f(a+ AX) Ax + f(a+2 Ax) Ax + ••• + f(a+(n-1) Ax) Ax n-1 n • SS- f(a+k•AX) AX = L f(a+(k-l)•Ax) Ax. 1c=o k=l Of course, as the number of rectangles is increased between a and b, the value of Ax gets smaller, and the value of Aa becomes a better approximation to the exact area 26 A exact· In other words, n-1 n A • lim L f(a+k•Ax) Ax "' lim L f(a+(k-1) Ax) Ax. exact n~o0 k=O n-..• k=l However, it is by working specific examples that student confidence in the fundamental theorem can be established. Specif-ic examples will require a knowledge of sigma notation properties (see table 1). The properties are usually proven using mathemati-cal induction in a precalculus course and can be repeated for emphasis in a lecture on this material. TABLE 1 PROPERTIES OF THE SIGMA NOTATION n }: c = en k=l n k = n(n + 1) 2 k=l 2 n k2- n(n + 1)(2n + 12 ~ - 6 n k3=M2 '2 k=l . 2 As a specific example, consider the area between the graph of f(x) = x2 + 1 and the x-axis on the interval l ,S. x ,S. 4 (see fig-ure 2). y 1' I FIGURE ~ --.. ~ <".IC: <".I;:: <".I,:; ~ '-" .__, + C\J C""\ -f' + • •• / I t::.x r.ic 4 --.. .... I ~ -4--27 /~f{x) = x2 + 1 t::.x = b-a - 4-1 1 n - -- = n n I I 1 1'1_: ! 111 ii 11:i \ i \_ A a 3 3 3 ( 3 3 f(l)(-) + f(l + -)(-) + ••• + f 1 + (n-1)-) (-) n n n n n 3 n-1 3 = - ":> f(l + k(-)) n M n Since f(x) 2 X + 1, then f(l + k(i)) (1 + 3k)2 + l n 2 + 6k + 9t2 .n 2 n 3 f 6k ~ 2 3 A = - ( 2 +- + ) = - ( a n K::O n n n )1-1 i' )I-/ Y 2 + i k + 1k L k 2 > (=-O n 1<:0 n J(.:o l {z(n-l) + 6(n-l)(n) + 9(n-l)(n)(2n-l)) n 2n 60 = 48n2 - 57n + 9 2n2 Now since A represents a 2 the approximate area under x + 1 from 1 to 4 with n rectangles, taking the limit as n approaches infinity gives the exact area. 2 4. = lim 48n - 57n + 9 = ~ nexact ~"° 2 n 2n J4 2 3 14 and (x + 1) dx = .!.._ + x 1 3 1 = 24. -Although this is just one example, it is strong evidence for the truth of the fundamental theorem. After several examples of this technique have been given with their integral counterpart, a formal proof of the fundamental theorem may be attempted. Once the fundamental theorem has been established and the students are convinced of its truth, the teacher could take the opportunity to show the double-integral representation of area and its equivalence to the single integral and the fundamental theorem (see figure 3). 28 - ~ y = f(x) ~ ::,·=f( x) ~ [r:x) dx .a b I a b FIGURE d l b rf(x) lb lf(x) ib a Jo dy dx = a y O dx = a f(x) dx. The single integral approach can be thought of as the in-scription of infinitely many rectangles under f(x) and above the x-axis on a_s: x .S: b. However, the double integral represents a summation of the area of infinitely many rectangles packed under f(x) and above the x-axis (see figure 4). 6x l::!,.y , v=z·-=-t~~:...c--~ " . ""Lt~:~",,:c; I iarcx) --~ ~- -: t-~~~~~;=tf t}tr y -- ~--FIGURE!± Each of the 'typical I rec tangles has a length Ii x and a width fi y. In fact, the logic is analogous to that used in the Riemann In te-gration approach. It is by summing the rectangle areas that we can approximate the exact area. 29 I . i I I 111 I I, 1l 1 !1 f._l, 1.,' i' \. A a 3 3 3 ( 3 3 f(l)(-) + f(l + -) (-) + ••• + f 1 + (n-1)-) (-) n n n n n 3 n-1 3 = - ">" f(l + k(-)) n ~ n Since f(x) 2 X + 1, then f(l + k(i)) (1 + 3k)2 + l n 2 + 6k + 9t2 .n 2 n A a =1 II-I 6k 91c2 3 'II-I 6 -'=1 9 'II-/ 2 L ( 2 +- + ~ ) = - ( L 2 + - ,2:_ k + ""2 L k ) n K::o n n n (""" n 1(:0 n l(:o 1 (zcn-1) + 6(n-l)(n) + 9(n-l)(n)(2n-l)) n 2n 6n = 48n2 - 57n + 9 2n 2 Now since Aa represents the approximate area under x + 1 from 1 to 4 with n rectangles, taking the limit as n approaches infinity gives the exact area. 2 4. = lim 48n - 57n + 9 = ~ r1exact ~oO 2 n 2n J4 2 3 14 and (x + 1) dx = .!...._ + x = 24. 1 3 1 -Although this is just one example, it is strong evidence for the truth of the fundamental theorem. After several examples of this technique have been given with their integral counterpart, a formal proof of the fundamental theorem may be attempted. Once the fundamental theorem has been established and the students are convinced of its truth, the teacher could take the opportunity to show the double-integral representation of area and its equivalence to the single integral and the fundamental theorem (see figure 3). 28 - ~ y = f(x) ~ y=f( x) ~ [,~x) dx .a b a b ~J l b rf(x) lb lf(x) rb a j O dy dx = a y O dx = Ja f (x) dx. The single integral approach can be thought of as the in-scription of infinitely many rectangles under f(x) and above the x-axis on a_s: x ,S: b. However, the double integral represents a summation of the area of infinitely many rectangles packed under f(x) and above the x-axis (see figure 4) • 6x /:ly '--/~. t:~~-~~2::~ ~-;..-~~:.~·:--:- r I y=f( x) --=--LT '----!±-t'::- t - - '·--· _,__ __ J.·p:.:+ -~ i : ~~ ~-~ /-~--::~ :L, ·' - . C FIGURE~ Each of the 'typical' rec tangles has a length fl x and a width fl y. In fact, the logic is analogous to that used in the Riemann Inte-gration approach. It is by summing the rectangle areas that we can approximate the exact area. 29 A a L ~t.yt:.x region EI: /lx fl y region For convenience let 6A = fly t:.x. Now intuition leads the student to believe the true result that t:.A1:!;0 LL t:.A ('b f(x) JS j a Jo dA ~ dy dx. The point here is that by developing the student's intuition one can build insight that can lead to understanding rather than memorization and frustration. It is after a concept is understood that formal proof should fit into advanced placement calculus. Rigor is important in mathematics and must not be totally omitted. However, a rigorous approach without a foundation in understanding is time wasted for most students. WORD PROCESSING: A TEACHER'S AIDE Roger W. Carlson Ohio University Athens, Ohio With the growing presence of microcomputers in colleges and schools, used both administratively and for computer-assisted instruction(CAI), their use outside the classroom by teachers becomes more common. Whether the micro is at home or at school, it can be of considerable assistance, especially when the amount of work to be done exceeds the time in which to do it. Many teachers, particularly in ma.thematics, are already acquainted with the value of computers for teaching ma.thematics and for student use in writing programs, entering formulas and solving problems. What is important, however, is to know that there are other microcomputer applications which .make the teacher's work faster, more accurate, more uniform and much easier. JO Some have argued that this large amount of teacher-talk is appro-priate for mathematics instruction, where students "must be told what to do." Others; however, have countered that mathematics should be more oriented toward student participation and discovery if mathematics is to be meaningful. Perhaps of even more importance are these ideas: Better teachers devote about half of the instructional time to lecture, demonstration, and discussion, and some-what less than half the time to individual seatwork for practice. Less effective teachers devote about one-fourth of the instructional time to lectures, etc., and over half the time to seatwork. (Evertsen et al., 1980b) This recent research evidence supports findings by Shipp and Deer and others in the 1960's. There appears to be a connection be-tween the amount of time devoted to developing mathematical ideas, and student achievement. What is the typical pattern of mathematics classes? Several recent studies have indicated that the following is the most usual plan: Opening activities - 5 minutes Checking/grading homework - 10 minutes Lecture/discussion - 10 minutes Seatwork - 20-30 minutes Closing activity - 5 minutes Good and Grouws (1979) have proposed an alternative model that has been demonstrated to promote achievement: Daily review - 8 minutes Development - 20 minutes Seatwork - 15 minutes Homework - daily Special reviews (of all content) - once a week With this model, a greater amount of time is spent on developing mathematical content. Moreover, it incorporates systematic re-view, an ingredient that has been missing from some mathematics .3 

Single and Double Integrals for Area in Advanced Placement Calculus

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Single and Double Integrals for Area in Advanced Placement Calculus

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