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04:31
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Q: Find the area under the curve of the given function using the limit sum.

buddhababe$$\lim_{n\rightarrow \infty}\sum_{k=1}^{n} {f(c_k)\triangle x_k}$$ If $y = x^{2} - 4x + 6$ from $[0,8]$, find the exact area under the parabola with the region being partitioned into four rectangles each of width 2.

limits summation area
buddhababe
buddhababe
I know that $\triangle x = 2$ and I get that the area is 12.
Masacroso
Masacroso
the question is not clear, you want to evaluate the area under $y$ in $[0,8]$ or the area of a Riemann sum of four rectangles?
buddhababe
buddhababe
I believe the professor would like the area of the Riemann sum of the four rectangles under the curve.
Masacroso
Masacroso
the Riemann sums are based in tagged partitions, so it value can be very different if you choose different tags. I assume that your professor (probably) wanted the Darboux upper sum or the Darboux lower sum of the defined partition.
buddhababe
buddhababe
Yes, so I'm looking for the Darboux lower sum.
Masacroso
Masacroso
04:31
Then use the definition of the Darboux lower sum, see here. I mean, what is your problem in this exercise?
buddhababe
buddhababe
What is $M_i$?
Masacroso
Masacroso
$M_i$ is the supremum of the function in the subinterval $i$. If the function is continuous then the supremum is the maximum of the function in this subinterval. But this is used to evaluate the Darboux upper sum, you want to evaluate the Darboux lower sum, right?
buddhababe
buddhababe
Yeah. I'm so confused.
Do you know if I can evaluate it as simply n^2 -4n + 6 as n approaches infinity?
Masacroso
Masacroso
No, I dont know why you want to to do that, but $$\lim_{n\to\infty}n^2-4n+6=\infty$$
buddhababe
buddhababe
Oh, okay.
So then n(n +1)
n(n+1)^2 - 4(n(n+1)) + 6
Masacroso
Masacroso
04:43
I dont understand what are you trying to do. I dont know exactly the context of your original question. My recommendation is that you look again what your professor was teaching the last days.
buddhababe
buddhababe
I'm just trying to find the area under the given curve!
Haha. Sorry for sounding sassy...
I know the function and the interval?
Masacroso
Masacroso
But this is a different thing that find some Darboux sum. This is the reason why your original question was not so clear. The area under the curve is found when the limit that you showed in the heading of your question, where $\Delta x_k\to 0$, is well defined for any tagged partition.
buddhababe
buddhababe
Oh, okay.
so from 0 to 8 there are 4 rectangles
Masacroso
Masacroso
the rectangles are defining a specific partition of the interval $[0,8]$, you said that the width of each subinterval is $2$ so $\Delta x_k=2$.
buddhababe
buddhababe
2, 4, 6, 8
are the subintervals
Yep! :)
Do I just integrate?
Masacroso
Masacroso
04:57
the integral is the area under the curve
buddhababe
buddhababe
okay, yeah
so if the professor was non-specific about lower/upper
integrate the function, evaluate
Masacroso
Masacroso
it is possible, yes. I dont know what is the context (the theory that your professor was teaching) to know how to answer your question.
buddhababe
buddhababe
lower/upper not even on the paper haha

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