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When this polar curve intersects with the $y$-axis, why is the value of the angle $fracpi2$?

Trying to plot these points in a polar coordinate systemPolar equation — find area under graph using double integralJust learned about the bell curve in statistics. How is calculus related to this curve?Integral of absolute value of X and area under the curve.Area under the curve described by θ=arDetermining bounds for polar areaFind the domain of the polar curve $r(theta)=2,cos2theta$Why does the integral for the arc length of a polar curve have the boundaries it has?Area of the figure bounded by curve, in polar coordinatespolar graphing, supposed range is between $0$ to $pi,$ but graph drawn range from $-pi/2$ to $pi/2$

0

1

$begingroup$

I'm currently learning area bounded by polar curve on Khan Academey, in one of the exercise, they asked

I don't really understand how the value for beta is obtained (or to be more precise, the principle behind how the value was obtained), could someone please expand the answer a little bit?

calculus polar-coordinates

share|cite|improve this question

edited Mar 21 at 5:15

Thor

asked Mar 21 at 5:04

ThorThor

26618

$endgroup$

add a comment | 

0

1

$begingroup$

I'm currently learning area bounded by polar curve on Khan Academey, in one of the exercise, they asked

I don't really understand how the value for beta is obtained (or to be more precise, the principle behind how the value was obtained), could someone please expand the answer a little bit?

calculus polar-coordinates

share|cite|improve this question

edited Mar 21 at 5:15

Thor

asked Mar 21 at 5:04

ThorThor

26618

$endgroup$

add a comment | 

$begingroup$

I'm currently learning area bounded by polar curve on Khan Academey, in one of the exercise, they asked

I don't really understand how the value for beta is obtained (or to be more precise, the principle behind how the value was obtained), could someone please expand the answer a little bit?

calculus polar-coordinates

share|cite|improve this question

edited Mar 21 at 5:15

Thor

asked Mar 21 at 5:04

ThorThor

26618

$endgroup$

share|cite|improve this question

edited Mar 21 at 5:15

Thor

asked Mar 21 at 5:04

ThorThor

26618

share|cite|improve this question

edited Mar 21 at 5:15

Thor

asked Mar 21 at 5:04

ThorThor

26618

asked Mar 21 at 5:04

ThorThor

26618

asked Mar 21 at 5:04

ThorThor

26618

1 Answer
1

active

oldest

votes

1

$begingroup$

The gist of it is that in polar coordinates $theta=fracpi2$ corresponds to the $y$-axis. So, if we want a point on the $y$-axis with the restriction that $0leqthetaleqpi$, then we must have $theta=fracpi2$.

Edit: The only systematic way to find the limits of integration in these cases is to solve for the relevant points of intersection (probably not what you wanted to hear/were looking for). But, this will get much easier with practice. It seems that your confusion is stemming from some difficulty or inexperience in thinking about polar coordinates. For example, this case is quite different from the case of Eulidean coordinates where we set two expressions equal and solve. In this case, we are not necessarily trying to solve for $r=0$, but rather identifying an angle that will satisfy the given condition. As far as recommended reading, most first semester calculus texts should discuss this topic. I learned calculus from Stewart's text and have also used it in teaching. You could probably get a used older edition on the cheap and I'd recommend it as a basic calculus text. Aside from that, perhaps try to read about polar coordinates online (wikipedia, online lecture notes, etc) and get some practice thinking in polar as opposed to Euclidean coordinates. I hope this long, somewhat rambling edit is of some help.

share|cite|improve this answer

edited Mar 21 at 5:55

answered Mar 21 at 5:20

Gary MoonGary Moon

92127

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks Gary for your help. Do you mind if I ask a follow up question. I am really confused about how to find the upper and lower bound when solving these type of equations, is there a systematic way to find the upper and lower bound? is there any reading you would recommend?
  $endgroup$
  – Thor
  Mar 21 at 5:25
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Thor I edited my answer to (attempt to) address your question.
  $endgroup$
  – Gary Moon
  Mar 21 at 5:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thank you so much for helping out! your answer will definitely help me go a long way.
  $endgroup$
  – Thor
  Mar 21 at 6:06
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm glad to hear it and it's my pleasure. I enjoy having the opportunity to talk with people about math.
  $endgroup$
  – Gary Moon
  Mar 21 at 6:12
  

  
  
  
  

add a comment | 

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1

$begingroup$

The gist of it is that in polar coordinates $theta=fracpi2$ corresponds to the $y$-axis. So, if we want a point on the $y$-axis with the restriction that $0leqthetaleqpi$, then we must have $theta=fracpi2$.

Edit: The only systematic way to find the limits of integration in these cases is to solve for the relevant points of intersection (probably not what you wanted to hear/were looking for). But, this will get much easier with practice. It seems that your confusion is stemming from some difficulty or inexperience in thinking about polar coordinates. For example, this case is quite different from the case of Eulidean coordinates where we set two expressions equal and solve. In this case, we are not necessarily trying to solve for $r=0$, but rather identifying an angle that will satisfy the given condition. As far as recommended reading, most first semester calculus texts should discuss this topic. I learned calculus from Stewart's text and have also used it in teaching. You could probably get a used older edition on the cheap and I'd recommend it as a basic calculus text. Aside from that, perhaps try to read about polar coordinates online (wikipedia, online lecture notes, etc) and get some practice thinking in polar as opposed to Euclidean coordinates. I hope this long, somewhat rambling edit is of some help.

share|cite|improve this answer

edited Mar 21 at 5:55

answered Mar 21 at 5:20

Gary MoonGary Moon

92127

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks Gary for your help. Do you mind if I ask a follow up question. I am really confused about how to find the upper and lower bound when solving these type of equations, is there a systematic way to find the upper and lower bound? is there any reading you would recommend?
  $endgroup$
  – Thor
  Mar 21 at 5:25
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Thor I edited my answer to (attempt to) address your question.
  $endgroup$
  – Gary Moon
  Mar 21 at 5:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thank you so much for helping out! your answer will definitely help me go a long way.
  $endgroup$
  – Thor
  Mar 21 at 6:06
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm glad to hear it and it's my pleasure. I enjoy having the opportunity to talk with people about math.
  $endgroup$
  – Gary Moon
  Mar 21 at 6:12
  

  
  
  
  

add a comment | 

1

$begingroup$

The gist of it is that in polar coordinates $theta=fracpi2$ corresponds to the $y$-axis. So, if we want a point on the $y$-axis with the restriction that $0leqthetaleqpi$, then we must have $theta=fracpi2$.

Edit: The only systematic way to find the limits of integration in these cases is to solve for the relevant points of intersection (probably not what you wanted to hear/were looking for). But, this will get much easier with practice. It seems that your confusion is stemming from some difficulty or inexperience in thinking about polar coordinates. For example, this case is quite different from the case of Eulidean coordinates where we set two expressions equal and solve. In this case, we are not necessarily trying to solve for $r=0$, but rather identifying an angle that will satisfy the given condition. As far as recommended reading, most first semester calculus texts should discuss this topic. I learned calculus from Stewart's text and have also used it in teaching. You could probably get a used older edition on the cheap and I'd recommend it as a basic calculus text. Aside from that, perhaps try to read about polar coordinates online (wikipedia, online lecture notes, etc) and get some practice thinking in polar as opposed to Euclidean coordinates. I hope this long, somewhat rambling edit is of some help.

share|cite|improve this answer

edited Mar 21 at 5:55

answered Mar 21 at 5:20

Gary MoonGary Moon

92127

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks Gary for your help. Do you mind if I ask a follow up question. I am really confused about how to find the upper and lower bound when solving these type of equations, is there a systematic way to find the upper and lower bound? is there any reading you would recommend?
  $endgroup$
  – Thor
  Mar 21 at 5:25
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Thor I edited my answer to (attempt to) address your question.
  $endgroup$
  – Gary Moon
  Mar 21 at 5:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thank you so much for helping out! your answer will definitely help me go a long way.
  $endgroup$
  – Thor
  Mar 21 at 6:06
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm glad to hear it and it's my pleasure. I enjoy having the opportunity to talk with people about math.
  $endgroup$
  – Gary Moon
  Mar 21 at 6:12
  

  
  
  
  

add a comment | 

$begingroup$

The gist of it is that in polar coordinates $theta=fracpi2$ corresponds to the $y$-axis. So, if we want a point on the $y$-axis with the restriction that $0leqthetaleqpi$, then we must have $theta=fracpi2$.

Edit: The only systematic way to find the limits of integration in these cases is to solve for the relevant points of intersection (probably not what you wanted to hear/were looking for). But, this will get much easier with practice. It seems that your confusion is stemming from some difficulty or inexperience in thinking about polar coordinates. For example, this case is quite different from the case of Eulidean coordinates where we set two expressions equal and solve. In this case, we are not necessarily trying to solve for $r=0$, but rather identifying an angle that will satisfy the given condition. As far as recommended reading, most first semester calculus texts should discuss this topic. I learned calculus from Stewart's text and have also used it in teaching. You could probably get a used older edition on the cheap and I'd recommend it as a basic calculus text. Aside from that, perhaps try to read about polar coordinates online (wikipedia, online lecture notes, etc) and get some practice thinking in polar as opposed to Euclidean coordinates. I hope this long, somewhat rambling edit is of some help.

share|cite|improve this answer

edited Mar 21 at 5:55

answered Mar 21 at 5:20

Gary MoonGary Moon

92127

$endgroup$

share|cite|improve this answer

edited Mar 21 at 5:55

answered Mar 21 at 5:20

Gary MoonGary Moon

92127

answered Mar 21 at 5:20

Gary MoonGary Moon

92127

answered Mar 21 at 5:20

Gary MoonGary Moon

92127