MAT 156Lab 6Topic 1 : Calculating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=We will use the fact that 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 is the area of a circle of radius 1 which is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=f:= x-> sqrt(1-x^2);plot(f(x), x=-1..1, scaling=constrained);Assignment Part 1: Compute the left-hand sums and right-hand sums with 10, 100, 1000 subintervals. Do not forget the student package.Assignment Part 2: What do you notice? Can you explain what you noticed? You may want to graph some left-hand sums and right-hand sums to explain your answer.As a result we do not get underestimates and overestimates for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= using the previous commands.Assignment Part 3: Find underestimates and overestimates for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=, using left-hand sums and right-hand sums with 5, 50, 500, 5000 on the intervals [-1,0] and [0,1].We can actually predict beforehand how large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== should be so that our underestimates and overestimates are within, say, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==. Recall that on the interval [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==], the difference |LHS(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==)-RHS(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==)|=|f(b)-f(a)|LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJkRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== for a monotone function. If we split [-1,0] into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== subintervals the error is less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JJm1mcmFjR0YkNigtRiM2JUYrLUYjNidGKy1GIzYlLUYsNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GQVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRksvJSlzdHJldGNoeUdGSy8lKnN5bW1ldHJpY0dGSy8lKGxhcmdlb3BHRksvJS5tb3ZhYmxlbGltaXRzR0ZLLyUnYWNjZW50R0ZLLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEiMEYnRkdGRy1GRDYtUSgmbWludXM7RidGR0ZJRkxGTkZQRlJGVEZWL0ZZUSwwLjIyMjIyMjJlbUYnL0ZmbkZkby1GIzYlRjpGQy1GaG42JC1GIzYlRistRiM2JC1GRDYtUSomdW1pbnVzMDtGJ0ZHRklGTEZORlBGUkZURlZGY29GZW8tRl1vNiRRIjFGJ0ZHRitGR0YrRistRiM2Iy1GLDYlUSJuRidGPUZALyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZecS8lKWJldmVsbGVkR0ZLLUZENi1RIj1GJ0ZHRklGTEZORlBGUkZURlYvRllRLDAuMjc3Nzc3OGVtRicvRmZuRmdxLUYyNigtRiM2I0ZhcEZkcEZpcEZccUZfcUZhcUYr. since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJkRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw===(b-a)/LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==. On the interval [0,1], if we split into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== subintervals, we have an error less than 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. So in the whole interval [-1,1] the error is less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y1USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRy8lKWJldmVsbGVkR1EmZmFsc2VGJw== and, since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= is twice the integral, the error in estimating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= is less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiNEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y1USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRy8lKWJldmVsbGVkR1EmZmFsc2VGJw==. If we want the error to be, say less than 0.001, we need to make LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JJm1mcmFjR0YkNigtRiM2Iy1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Iy1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRjtRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZMLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbW9HRiQ2LVEiPEYnRjovJSZmZW5jZUdGUS8lKnNlcGFyYXRvckdGUS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zgby1GNzYkUSYwLjAwMUYnRjpGKw==, which gives 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, i.e., n>4000. This explains why with 5000 subintervals we had gotten the first 3 decimals correct.Assignment Part 4: How large do you need to take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUyNTdGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=, so that you can compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRMkNoYXJhY3RlclN0eWxlMjU5RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn with an error less than 0.0000001?Unfortunately, this number of subintervals is too large to work out with Maple. So we resort to the trapezoid rule and the midpoint rule.Assignment Part 5: Compute TRAP(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==), MID(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==), with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw===10, 100, 1000 on the whole interval [-1,1]. Which ones give underestimates and which ones give overestimates of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRMkNoYXJhY3RlclN0eWxlMjYyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn? Why? What is the relation of TRAP(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==) with LHS(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==)?To compute more accurately, we can use Simpson's rule: SIM(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==)= 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 . There is a Maple command for Simpson's rule:simpson(f(x), x=lowerlimit..upperlimit, 2*number of subintervals)Assignment Part 6 : Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRMkNoYXJhY3RlclN0eWxlMjcxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn using Simpson's rule with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUyNzNGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic==10, 100, 1000,10000 subintervals. Optional: Verify your answers using the formula for Simpson's rule.Topic 2 : Substitution in MapleAlthough Maple can often evaluate integrals directly it is useful to be able to use the substitution technique.Example: Compute 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 substitutionE:=Int((x^2+1)^4*x, x);The first step is to substitute 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:=changevar(u=x^2+1, E, u);The second step is to evaluate this integralV:=value(F);The last step is to substitute x back into the answerS:=subs(u=x^2+1, V);For comparison, we evaluate directly value(E);This seems different from S but they are not - we expand S to check.expand(S, x);We still notice that there is a difference: the constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUYxNiRRIzEwRidGNC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGQS8lKWJldmVsbGVkR1EmZmFsc2VGJw==. This is so, because in integration Maple does not care about constants. The most general antiderivative is any of the previous expressions +C.One problem is that Maple does not tell us which substitution to use. So the choice is ours and this is where our work lies.Assignment Part 7 (A) Find 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 directly by expanding 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 (There is a Maple command called expand that you can use.) and then just integrating the terms. (B) Find 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 using the substitution 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What do you conclude?Assignment Part 8 Find 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 using a substitution. Check your answer with the value command.
Assignment Part 9 Find 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 using a substitution. Check your answer with the value command.