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Chapter 5 5

Chapter 5 5 - SECTION 5.1 AREAS AND DISTANCES 437 At the...

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Unformatted text preview: SECTION 5.1 AREAS AND DISTANCES 437 At the end of this procedure, (DELTA_X) . (SUM) is equal to the answer we are looking for. We ﬁnd that 71' 30 27r 171' N 7r 50 £19993 R102ﬁ2081n(1—0>~ 1.9835 Rgozg—OZsin<3—0> 1.9982 andR50=— 5OZsin<5—0> . . :1 It appears that the exact area is 2. Shown below is program SUMRIGHT and its output from a TI-83 Plus calculator. To generalize the program. we have input (rather than assigned) values for Xmin. Xmax. and N. Also. the function. sin at. is assigned to Y1. enabling us to evaluate any right sum merely by changing Y1 and running the program. PPQMSUMRIEHT Hm1n=?B Hmax=?n H=?18 1.98352353? Done PRDERHM=SUHRIEHT =B+S _ :Prnmpt Hm1n :Prompt Hmax =PFDMPt H_ =(Hmax—HM1HJXH+D =Hm1n+D+R :FOP'L'I:1:-H:I =S+V1(R)+S =R+D+R =End :D\$S+2 =DlsP 2 . We can use the algorithm from Exercise 7 with X_MIN : 1. X_MAX : 2. and 1 / (RIGHT‘ENDPOINT)2 instead 1 10 1 f S' RIGHT ENDPOINT in Ste 221. We ﬁnd that R : — _ RV.» 0.4640. 0 1n( _ ) p 10 10; (1+i/10)2 1 30 50 1 1 1 R : — —— % 0.4877. and R — — —— N 0.4926. It a ears that the exact area ”:m;a+mw “ 50;(1+mm w ~ 1 152. . In Maple. we have to perform a number of steps before getting a numerical answer. After loading the student package [command: with (student) ;] we use the command 1eft_sum: =leftsum (xA (1/2) ,x=1. .4 , 10 [or 3 O. or 50]) ; which gives us the expression in summation notation. To get a numerical approximation to the sum. we use evalf (left_sum) ; . Mathematica does not have a special command for these sums. so we must type them in manually. For example, the ﬁrst left sum is given by (3/10) *Sum[Sqrt [1+3 (i—l) /10] , {1, l, 10}] , and we use theNcommand on the resulting output to get a numerical approximation. In Derive. we use the L3 FT_RIEMANN command to get the left sums. but must deﬁne the right sums ourselves. (We can deﬁne a new function using LEFT_RIEMANN with k ranging from 1 to n instead of from 0 to n ~ 1.) (a) With f(ac) : ﬂ. 1 S m S 4. the left sums are of the form L,L = E Z + 30 ~ 1). Speciﬁcally. n .1 n L10 % 4.5148. L30 % 4.6165, and L50 a“ 4.6366. The right sums are of the form Rn : g E 1 + a. i:1 n Speciﬁcally. R10 x 4.8148. R30 % 4.7165. and R50 x 4.6966. ...
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