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butterflydreamer
 one year ago
Question >
http://prntscr.com/7tc26v
I just need a hint on how to approach Part (ii) :) Thank you !
butterflydreamer
 one year ago
Question > http://prntscr.com/7tc26v I just need a hint on how to approach Part (ii) :) Thank you !

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0Simply multiply the width by corresponding height to get the area of \(r\)th rectangle. Let \(f(x) = x^3+x\) and notice below : Width of each rectangle is given by \(\dfrac{10}{n} = \dfrac{1}{n}\) Height of \(r\)th rectangle is given by \(f(\dfrac{1}{n})\).

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1if you label the rectangles with r=1 r=2 .... r =n1 (instead of the a1 a2 you have drawn on the diagram) you can see that the x value of the left side of th erectangle is r/n for each rectangle SO using the equation for y you can work out the HEIGHT of the rectangle Height = (r/n)^3 + r/n The WIDTH of each rectangle is 1/n SO Multiply the height by the width to get the area of each rectangle You will see the form of the answer begin to appear when you write this expression. The total are is simply the sum from r = 1 to r = n1 of all those areas, which leads tot the answer shown....

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0typo fixed : Let \(f(x) = x^3+x\) and notice below : Width of each rectangle is given by \(\dfrac{10}{n} = \dfrac{1}{n}\) Height of \(r\)th rectangle is given by \(f(\dfrac{\color{red}{r}}{n})\).

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1\[\therefore A_n =\frac{ 1 }{ n^4 } \sum_{r = 1}^{n1}r^3 + \frac{ 1 }{ n^2 }\sum_{r=1}^{n1}r\]oh.. so it will be something like this...? \[A_1 = \frac{ 1 }{ n } ( \frac{ 1 }{ n^3 } + \frac{ 1 }{ n }) = \frac{ 1 }{ n^4 } + \frac{ 1 }{ n^2 }\] \[A_2 = \frac{ 1 }{ n } (\frac{ 8 }{ n^3 } + \frac{ 2 }{ n }) = \frac{ 8 }{ n^4 } + \frac{ 2 }{ n^2 }\] \[A_3 = \frac{ 1 }{ n } ( \frac{ 27 }{ n^3} + \frac{ 3 }{ n }) = \frac{ 27 }{ n^4 } +\frac{ 3 }{ n^2 }\] . . . \[A _{n1} = \frac{ 1 }{ n } \left[ \frac{ (n1)^3 }{ n^3 }+\frac{ n1 }{ n } \right]= \frac{ (n1)^3 }{n^4 }+\frac{ n1 }{ n^2 }\] \[A_1 + A_2 + ...+ A _{n1} = \frac{ 1 }{ n^4 } + \frac{ 8 }{ n^4 } + \frac{ 27 }{ n^4 }+...+\frac{ (n1)^3 }{ n^4}+ \frac{ 1 }{ n^2 }+\frac{ 2 }{ n^2 }+\frac{ 3 }{ n^2}+...+ \frac{ n1 }{ n^2}\] \[A_n = \frac{ 1 }{ n^4 } \left[ 1 + 8 + 27 + ... + (n1)^3\right]+ \frac{ 1 }{ n^2 }\left[ 1 + 2+...+(n1) \right]\]

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1oops. I made a typo LOL. The first line " therefore .." is meant to be at the end xD

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1yes  but you do not help by putting the numbers in ]

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1putting the numbers in ?

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1\[area _{n} = \frac{ 1 }{ n }( \frac{ r ^{3} }{ n ^{3} } \frac{ r }{ n })\]

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1hmm.............

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1you have put the numbers 1,2, 3 etc for r but oyu just need to leave it in terms of r and n expand the above  you will see the answer form appear

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1ahh okay. So what i did was unnecessary?

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1^^ typo in formula should be + not 

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1just distribute the expression I wrote above...

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1xD i know i know. I'm just saying.. like using the method you demonstrated, i'd be able to show the expression faster right?

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1it's not just faster  the numbers do not appear in the answer  so you are not going to need them to derive the answer after you have distributed  then you need to write the 'summation' equation remember n is a constant  so it can be taken outside the summation....

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1another typo in formula it should be area r not area n

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1so we're using: \[A_r = \frac{ 1 }{ n } ( \frac{ 1 }{ n^3 } + \frac{ 1 }{ n }) \] ? And i just plug in r = 1, 2 , 3 ..... (leaving it in terms of r and n )and eventually i'll reach the summation equation after expanding and everything :) Okay thank you!

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1oh typo *\[A_r = \frac{ 1 }{ n } (\frac{ r^3 }{ n^3 }+\frac{ r }{ n })\]

phi
 one year ago
Best ResponseYou've already chosen the best response.0once you have \[ A_n =\frac{ 1 }{ n^4 } \sum_{r = 1}^{n1}r^3 + \frac{ 1 }{ n^2 }\sum_{r=1}^{n1}r \] your are done. the next step would be to use "closed forms" to replace the summations

phi
 one year ago
Best ResponseYou've already chosen the best response.0The first summation is given by Part 1 the second is well known

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1NO multiply out the formual as I wrote it (with corrections) You NEVER need to put 1,2, etc into it  it does not ask you to work out abn answer  it asks oyu to prove the expression given at the bottom of the section. The expression above when multiplied gives you an expression for the area of ONE rectangle you then need to put this int a 'summation' form to show that the expression is true.

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1I reiterate: You are not asked to work out a value for the area or summation Since you do not know what n is you CANNOT Just derive the summation formula using the method above do you want to write your expansion of the expression here?

butterflydreamer
 one year ago
Best ResponseYou've already chosen the best response.1\[so. A_r = \frac{ 1 }{ n } (\frac{ r^3 }{ n^3 } + \frac{ r }{ n }) = \frac{ r^3 }{ n^4 } + \frac{ r }{ n^2 }\]

MrNood
 one year ago
Best ResponseYou've already chosen the best response.1OK good now  that is the area of 1 rectangle so the area of all the rectangles is the summation of that expression from r=1 to rn1 if you write that out in the notation used in the answer you will see that you have the right answer remember 2 things in simplifying: sum (A+B) = sum (A) + sum(B) and sum (A/n) = 1/n sum(A) if n is a constant use these and you will derive the answer given..
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