Using sigma sums to estimate the area under a curve

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Homework Statement



My apologies in advance for the messiness of the equations; the computers available to us do not correctly process the LaTex code.

I am tasked with estimating the area under the curve f(x)=x2+1 on the interval [0,2] using 16 partitions. Online calculators and my graphing calculator return a result of (14/3) but when I try to do the math by hand, I end up with 18.92188, and I can not find my error.

Homework Equations



[nƩ(i=1)] i2 = (n(n+1)(2n+1))/6
[nƩ(i=1)] f(x) = (n)Ʃ(i=1) f(xi)(Δx)

The Attempt at a Solution



From the given information:
Δx = (2-0)/n = 2/n
xi = 0 + (Δx)*i = 2i/n

[nƩ(i=1)] x2 + 1
[nƩ(i=1)] ((2i/n)2+1)*(2/n)

Moving constants to the left yields:

(2/n) [nƩ(i=1)] (22/n2)i2 + [nƩ(i=1)] 1
(2/n)(22/n2) [nƩ(i=1)] i2 + n

Converting the Ʃ equation to purely terms of n:

((8/n3)*(n(n+1)(2n+1))/6) + n
((8/n3)*((2n3+3n2+n)/6) + n
((16n3+24n2+8n)/6n3) + (6n4/6n3)

Add and factor out a 2n:

(3n3+8n2+12n+4)/(3n2)

Referring to the problem's instructions to use 16 partitions, I substitute 16 for n and I get 18.92188. Where did I go wrong?
 

Answers and Replies

  • #2
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You have multiple mistakes that fall under the banner of basic algebra. It's fairly common in my experience for calculus students to do just fine on the calculus part of the problem, but mess up the "easy" stuff, so you're not alone. I would advise that you work through it again, but maybe take a little more care to make good use of parentheses. Maybe take a few more steps to get the job done and try to identify the principle that applies to each step. Also, you might find that you make fewer mistakes if you don't try to mix concepts into the same step; i.e. don't use properties of/formulas related to sigma notation (new math to you) while trying to factor, distribute, etc. (old math); until you are more comfortable with the new stuff.

Presumably you know that as n→∞, the expression for the area that you get in terms of n should get closer to the actual area under the curve (since this function is positive on [0,2]). This gives you a way to check if your expression is reasonable; does the limit exist? is it a reasonable number?
 

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