Parametric Equations Watch

φM23
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#1
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I'm struggling to understand some of part d on question 3 in this paper:

http://www.madasmaths.com/archive/iy...apers/c4_y.pdf

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

http://www.madasmaths.com/archive/iy..._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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TeeEm
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(Original post by PhyM23)
I'm struggling to understand some of part d on question 3 in this paper:

http://www.madasmaths.com/archive/iy...apers/c4_y.pdf

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

http://www.madasmaths.com/archive/iy..._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?
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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.
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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)
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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
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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?
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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.
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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
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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?
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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
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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?
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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?
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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/Class.../ParaArea.aspx
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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/Class.../ParaArea.aspx
The answer will be the same whether you flip them or not ... It is done for convenience
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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?
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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
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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?
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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
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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?
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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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