precal
  • precal
Need help understand a definite integral using the limit of sums
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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precal
  • precal
\[\int\limits_{3}^{7}\left( 4x+6 \right)dx\]
precal
  • precal
I know how to do this problem but I am being ask to do it with summations and the limit of sums
precal
  • precal
\[ \lim_{n \rightarrow \infty} \sum_{n=1}^{\infty}f \left( x _{i} \right)\Delta x\]

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precal
  • precal
\[\lim_{n \rightarrow \infty}\sum_{i=1}^{n} \left[ 4\left( 3+\frac{4i}{n} \right)+6 \right]\frac{4}{n}\]
precal
  • precal
I do not understand the above statement
precal
  • precal
|dw:1332870959725:dw|
precal
  • precal
|dw:1332871012577:dw|
precal
  • precal
I don't know how they got \[(3+ \frac{4i}{n})\]
amistre64
  • amistre64
\[\int\limits_{3}^{7}\left( 4x+6 \right)dx\to\ \sum_{x=3}^{7}(4x+6)\Delta x\] where delta x is the increment value of yout interval
precal
  • precal
how did they get 4/n to be delta x?
amistre64
  • amistre64
dunno, im not them. If i had a look at the material you are using or something similar i might be able to decipher something tho
amistre64
  • amistre64
the interval from 3 to 7 is 4 units long; so maybe n is the value that they are slicing up that interval into
precal
  • precal
I still don't see \[4(3+\frac{4i}{n})+6\]
precal
  • precal
I can scan an example from my book would that help you? I have to use the right Riemann sum
amistre64
  • amistre64
it would give me something to go by other than taking a blind stab at it :) hopefully
amistre64
  • amistre64
theyve replaced x by that (....) somehow
amistre64
  • amistre64
i think i see a reasoning
amistre64
  • amistre64
the interval of x is from 3 to 7 so it starts at 3 and moves at most 4 units x = 3+ 4/n as n counts out from 1 to how ever many times we are subdividing this interval
amistre64
  • amistre64
4*i/n is taking slices of the interval that are 1/n th thick
amistre64
  • amistre64
when we have all the i/n pieces we have a whole interval n/n x = 3+4*0, 3+4*1/100, 3+4*2/100, ... , 3+4*100/100 if i counts from 0 to n and n = 100
precal
  • precal
ok give me a minute to scan and attach
precal
  • precal
1 Attachment
amistre64
  • amistre64
if n = 4; then i = 0 to 4 x = 3+4*0/4 , 3+4*1/4, 3+4*2/4, 3+4*3/4, 3+4*4/4 for each increment
precal
  • precal
1 Attachment
precal
  • precal
ok I have attached the problem from my solution manual it is problem 5.2.47 and I have attached the example from my book
amistre64
  • amistre64
yeah, that matches what I came up with while a stabbed at it :)
precal
  • precal
so delta x is the slices and f(xi) is the function with xi being substituted
amistre64
  • amistre64
|dw:1332877785490:dw|
amistre64
  • amistre64
that sounds about right yes
amistre64
  • amistre64
delta x = the width of a slice
precal
  • precal
I am so glad that there are more efficient methods for evaluating definite integrals.
amistre64
  • amistre64
there are n slices that keep adding up to the total interval from "s"tart to "f"inish by increments of "i"
amistre64
  • amistre64
me too, but its good to know how they are conceived at least :)
precal
  • precal
Thanks I think I can now better explain this to my relative.
amistre64
  • amistre64
your welcome ;)

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