Integral Question. See below,

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Integral Question. See below,

Mathematics
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O_o .
If \[\int\limits_{0}^{6} (x^2-2x+2)dx\] is approximated by 3 inscribed rectangles of equal width, then the approximation is... 1) 24 B) 26 C) 28 D)76 E) 48
sorry typed something wrong. and that 1 is supposed to be an A

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so i got the integrated area to be 48, so answers either a b or c
Wait.
Wait, what kind of rectangles? Left, right, or midpoint?
the problem doesnt say. i'm assuming left
cuz it says inscribed
Well you are right. It's either A, B or C :D . Now can you do it out? :) .
\[\Delta(x)=\frac{ 6-0 }{ 3 }\]
i'm confused my friend says to use riemann sum. what's the formula for that?
\[\Delta(x) = 2\]
In this particular example, a Riemann sum is basically the sum of the individual rectangles.
When you learn about definite intergals you will have a good idea about what they actually mean.
is it \[\frac{ b-a }{ n } \sum_{i = 1}^{n} f(x) \] ??
No wait, that's right.
So yeah you get the idea. It's the sum of all the rectangles multiples by the change in x,(i.e the width) .
that's what i did but i got 28 and the answer's 26
Then you have to use diffrent kinds of rectangles because your answer is correct.
26 or 28 ?
It's 26.
But if you use left rectangles then it's 28.
why? sorry can u explain?
oh.
so right hand it's 26 or..?
So this is a very stupid question because it dosen't specify the type of rectangles.
Well right hand is 76 actually.
But it dosen't specify what type of rectangle. Left, right or midpoint.
where'd the 26 come from? midpoint?
No, midpoint is 46.
But shall I explain left and right rectangles?
If you use left, you get 28. If you use right you get 76.
Shall I explain?
wait so where did the 26 come from. please explain. i'm so lost. :(
26 isn't the correct answer.
cuz this other person used 2(f(1)+f(2)+f(4)) and got 26 but i don't understand where the f(1) came from...
oh.
Wait hold on.
Yeah, 28 should be correct. I disagree with the answer.
If you use 3 subintervals you CANNOT get 26 in anyway.
The formula would be (2)(f(0)+f(2)+f(4))
yeah i did that... and got 28
Which is absolutely correct.
It CANNOT be 26 if you use left rectangles.
if you use right rectangles it's (2)(f(2)+f(4)+f(6)) which is 76.
yeah i got 28 and 76 but i chose 28. so my teacher is wrong. hmph.
Yep, you teacher is wrong.
thanks for clarifying. :)
In reality the exact value of that integral is 48 but you will learn that later.
yeah i know. i calculated it.
Kk.
Good 'ol Riemann's screwing with everybody's head.
I just hope I don't have to do it after my first year in uni :/ . AP calc helped so much though XD .
ohhhhhhh i think i got it they used average value
LOL
$$A\simeq2(2^2-2(2)+2+4^2-2(4)+2+6^2-2(6)+2)\\\ \ \ =2(4-4+2+16-8+2+36-12+2)\\\ \ \ =2(38)=76$$ (using right end-points)$$A\simeq2(0^2-2(0)+2+2^2-2(2)+2+4^2-2(4)+2)=2(2+4-4+2+16-8+2)\\\ \ \ =2(14)=28$$ (using left end-points)$$A\simeq2(1^2-2(1)+2+3^2-2(3)+2+5^2-2(5)+2)\\\ \ \ =2(1-2+2+9-6+2+25-20+2)\\\ \ \ =2(13)=26$$ (using midpoints)
^ Reason why they they should specify the question.
my friend just assumed that inscribed meant left end points
@oldrin.bataku : Please don't just give the answer next time. The point of this place is to help other people learn. Not give them the answer.
@jennychan12 : That's a flase assumption.
false*
@oldrin.bataku : If you use midpoints, it's 46 not 26.
Lol Oldrin and i have been spending plenty of time explaining problems to different users. I would give him a break xD. I was bored and decided to use this site for the first time today and, oddly enough, I can't stop attempting to solve problems. Probably because I don't want to forget my math whilst im on winter break.
I know but it's just that's it's against the TOS so I try not to. I make that mistake too. It's cool ^_^ .
@Dido525 you're right, 2(5) = 10 so it's actually 2(23) = 46 for the approximate area.
If it requires inscribed rectangles then the approximation must use left end-point... mid- and right- will not be entirely under the parabola.
That's why we have the trapozoidal rule ;) . Ans simpsons rule :P .
and*
But yes @oldrin.bataku is correct. If they MUST be inscribed then you MUSt use left rectangles.
MUST*
@Dido525 if you go into a Scientific Computation course you'll learn about plenty more numerical methods e.g. Runge-Kutta or Adams-Moulton :-p Euler's and Verlet are really common for physics problems.
Conversation is too much for this PoSci major xD.
I looked at Runge-kutta but it makes no sense XD . I am going much deeper into math next year so hopefully then :) .
Sorry that we turned this page into a discussion @jennychan12 .
it's aiight.
Nobody drew a damn diagram :P
Sigh...
Diagram was drawn in the other thread.
^ haha yes
|dw:1355899568667:dw|
Something like that.
that's the same thread as this one...
LOL
|dw:1355899651047:dw| That would be incorrect since it does not enclose the curve.
yeah i got really confused and tried to do another one to resolve my confusion... :D cuz that other person made me confused
|dw:1355899646424:dw|
|dw:1355899717852:dw|
If you use midpoints it still dosen't enclose the curve fully.
Inscribed rectangles *should* mean rectangles within the curve, but it sounds like that didn't work, so the damn question is broken.
Damn question.
It's is 28 though without a doubt.
More than likely your teacher made an error.
yeah
inscribed rectangles would give 26, but i feel like that was already covered in this mess somewhere
You haven't done "work" problems in your calc class yet have you?
Wait, why isnt the answer 26? 3 inscribed rectangles = area of 26.
what do u mean? kinetic energy? i do that in physics....
@agent0smith can u explain???
there are "work" problems you use calculus for. Pray that you do not have to do them xD
If you use 3 inscribed rectangles it's 28 not 26.
The previous thread came up with 26. this one says 28. let's meet half way and say it's 27...oh wait.
lol
The height of the first is 1, length is 2 height of the second is 2, length is 2 height of the third is 10, length is 2. 2+4+20
How'd you get 28, @Dido525
left endpoint is 28
=(2) * ( f(0) + f(2) + f(4) ) = (2) * ( 2 + 2 + 10 ) = 28
Inscribed have to be all within the curve, no edges jutting out: it's not f(0), it's f(1), the height of the first rectangle.
That is not an inscribed rectangle.
that's what that other person did in that other thread 2(f(1)+f(2)+f(4)) = 26 but why would be f(1)??? where does the 1 even come from?
But f(0) is inscribed.
I'll draw it jenny. One sec.
I want to see this too.
dido, look at the graph: http://www.google.com/search?q=x%5E2-2x%2B2&aq=f&oq=x%5E2-2x%2B2&sugexp=chrome,mod=18&sourceid=chrome&ie=UTF-8 and then look where a rectangle at f(0) would fall... it's not inscribed. it goes over the curve.
Isn't that the same thing as inscribed?
Inscribed meaning within.
No, inscribed means it falls completely within the curve, no jutting out over it.
okay. i understand that f(1) = 1 but.... oh wait. i see where you're going with this....
|dw:1355900439887:dw|
But rectangles will never cover a curve fully, it's impossible.
Like, it covers it fully but it's an overestimate.
Ohh shoot! O_o .
Those are inscribed rectangles, they underestimate the area greatly of course.
Ahh, NOW this makes sense.
hahah :)
But if it's an underestimate, it's not inscribed is it?
ok yeah i see it too.
I googled inscribed rectangles just to be sure.
Wait, why did you do a midpoint for the first rectangle?
Instead of a left one?
Because that's the only way to have it inscribed. http://en.wikipedia.org/wiki/Inscribed_figure
NOW this all makes sense :D .
Inscribed rectangles can be either left or right endpoint
or midpoint
@jennychan12 you see how it's 26 now? the height of the first is f(1), next is f(2), then f(4)
I would give you my medal if I could :P .
haha :) this whole confusion was all over the meaning of the word inscribed :D
Agreed! :D . But hey, this might be n my final so it was good for me too :) .
haha good :) hopefully this clears things up Jenny, before you went and told your teacher he/she was wrong because a bunch of people on the internet said so! :D
She is offline :/ .
@jada1701877 : You should have given that to agent.
thanks so much! and yeah, i went off at like 11 cuz i went to sleep. :)

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