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.
[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
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:
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?