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Parametric Equations

I'm struggling to understand some of part d on question 3 in this paper:

http://www.madasmaths.com/archive/iygb_practice_papers/c4_practice_papers/c4_y.pdf

I have looked at the solutions but I'm still confused:

http://www.madasmaths.com/archive/iygb_practice_papers/c4_practice_papers/c4_y_solutions.pdf

I don't get why you have to find the direction of the curve, and why the curve is being integrated between A and B instead of A and E to find the area.

Thanks for any help!

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Reply 1
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TeeEm
19
Original post by PhyM23
I'm struggling to understand some of part d on question 3 in this paper:

http://www.madasmaths.com/archive/iygb_practice_papers/c4_practice_papers/c4_y.pdf

I have looked at the solutions but I'm still confused:

http://www.madasmaths.com/archive/iygb_practice_papers/c4_practice_papers/c4_y_solutions.pdf

I don't get why you have to find the direction of the curve, and why the curve is being integrated between A and B instead of A and E to find the area.

Thanks for any help!


Do you realize this is a very hard paper only to be used as extension?
Reply 2
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φM23
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Original post by TeeEm
Do you realize this is a very hard paper only to be used as extension?


I do yes - I want to tackle harder questions to really test my understanding of the material to make completing the normal questions easier.
Reply 3
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TeeEm
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Original post by PhyM23
I do yes - I want to tackle harder questions to really test my understanding of the material to make completing the normal questions easier.



I need to look at the question and the solution.
I hope I can do it
(I have not revised parametrics)
Reply 4
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φM23
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Original post by TeeEm
I need to look at the question and the solution.
I hope I can do it
(I have not revised parametrics)


Thank you very much. I really appreciate your help
Reply 5
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TeeEm
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Original post by PhyM23
Thank you very much. I really appreciate your help


the direction of the curve, as theta increases is marked in the diagram at the beginning.
The area is swept in Cartesian from left to right, hence the limits


or if this further does not answer your question
do it from
theta = 2pi/3 to theta = zero
then subtract
from theta = -2pi/3 to theta =0


Any good?
Reply 6
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φM23
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Original post by TeeEm
the direction of the curve, as theta increases is marked in the diagram at the beginning.
The area is swept in Cartesian from left to right, hence the limits


or if this further does not answer your question
do it from
theta = 2pi/3 to theta = zero
then subtract
from theta = -2pi/3 to theta =0


Any good?


I'm sorry but I am still confused

Why isn't the area just equal to the integral between theta=0 and theta=2pi/3?
What do you mean by 'the area is swept left to right'
Why does the direction of the curve matter?
Where did the -2pi/3 come from in the second part of your answer?

I apologise if some/all of these questions are stupid. This question seems to contain everything about PEs that confuses me, so understanding this fully would help me greatly.
Reply 7
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TeeEm
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Original post by PhyM23
I'm sorry but I am still confused

Why isn't the area just equal to the integral between theta=0 and theta=2pi/3?
What do you mean by 'the area is swept left to right'
Why does the direction of the curve matter?
Where did the -2pi/3 come from in the second part of your answer?

I apologise if some/all of these questions are stupid. This question seems to contain everything about PEs that confuses me, so understanding this fully would help me greatly.


To find the area between the curve and the x axis you need to integrate from left to right so when it come to the limits from the smaller x to the larger x
Reply 8
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φM23
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Original post by TeeEm
To find the area between the curve and the x axis you need to integrate from left to right so when it come to the limits from the smaller x to the larger x


Why do you integrate from left to right? I thought you found the area with the greater cartesian limit on the top of the integral and the smaller limit on the bottom. Does it depend on the circumstance?
Reply 9
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TeeEm
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Original post by PhyM23
Why do you integrate from left to right? I thought you found the area with the greater cartesian limit on the top of the integral and the smaller limit on the bottom. Does it depend on the circumstance?


So we agree! Cartesian limits!
But these are the parametric limits which match these Cartesian limits
Reply 10
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φM23
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Original post by TeeEm
So we agree! Cartesian limits!
But these are the parametric limits which match these Cartesian limits


Oh I see in the solutions the parametric limits were initially the other way round to what they are in the answer. But why were they flipped?
Reply 11
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TeeEm
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Original post by PhyM23
Oh I see in the solutions the parametric limits were initially the other way round to what they are in the answer. But why were they flipped?


Do you know that if you reverse the limits you generate a minus?
Reply 12
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φM23
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Original post by TeeEm
Do you know that if you reverse the limits you generate a minus?


I do yes. So did you flip them because you saw that the answer would be negative if you didn't? Is this something you just have to spot earlier on or is there a particular way to tell. Does it have something to do with the explanation on this page in the blue boxes?:

http://tutorial.math.lamar.edu/Classes/CalcII/ParaArea.aspx
Reply 13
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TeeEm
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Original post by PhyM23
I do yes. So did you flip them because you saw that the answer would be negative if you didn't? Is this something you just have to spot earlier on or is there a particular way to tell. Does it have something to do with the explanation on this page in the blue boxes?:

http://tutorial.math.lamar.edu/Classes/CalcII/ParaArea.aspx


The answer will be the same whether you flip them or not ... It is done for convenience
Reply 14
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φM23
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Original post by TeeEm
The answer will be the same whether you flip them or not ... It is done for convenience


Ah okay that's understandable.

Please may you clarify why you mentioned in your solutions the direction of the curve? I.e. what significance does drawing arrows on the curve have?
Reply 15
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TeeEm
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Original post by PhyM23
Ah okay that's understandable.

Please may you clarify why you mentioned in your solutions the direction of the curve? I.e. what significance does drawing arrows on the curve have?


A curve in parametric has direction, which you get as the parameter increases.
No major significance but it shows how theta changes
Reply 16
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φM23
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Original post by TeeEm
A curve in parametric has direction, which you get as the parameter increases.
No major significance but it shows how theta changes


Ah okay. Why does the integral given as the answer only work out the area of the top part of the curve and not the bottom half of it as well?
Reply 17
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TeeEm
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Original post by PhyM23
Ah okay. Why does the integral given as the answer only work out the area of the top part of the curve and not the bottom half of it as well?


because these limits are associated with the top half
Reply 18
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φM23
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Original post by TeeEm
because these limits are associated with the top half


But isn't the bottom half also between these two limits?
Reply 19
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TeeEm
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Original post by PhyM23
But isn't the bottom half also between these two limits?


bottom is

from 2pi/3 to 5pi/3

or

-4pi/3 to -pi/3

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