I am trying to change the order of integration of a problem and I am getting confused at a point












0














My math book has a question of changing order of the following integral:



$$int_0^aint_{sqrt{ax-x^2}}^{sqrt{ax}} f(x,y),dx,dy.$$



So basically I have to change $dxdy$ into $dydx$ and their respective limits.



This is an example which is provided in my math textbook related to the problem. (For showing you diagram and make it easier for me to explain my confusion further.)



1]



Image Link



So in my problem, while changing the order of integration, the resultant changed integral order is coming in 3 terms (say, IOnfoot1+IOnfoot2+IOnfoot3). I1 is for OLC, I2 is for AMC and I3 is for BLCM.(Please refer to the diagram, which I shared in the form of Image)



While changing the circle, i.e. $y=sqrt{ax-x^2}$ into form of $x$, we get $x=frac{1}{2}left(apmsqrt{a^2-4y^2}right)$.



I tried to solve it ahead but I stucked on a problem of deciding in ± which to choose to get higher (upper) limit and to get lower limit. But in the answersheet of my Textbook, the following is given:-



In OLC the limit of y changes into [ from y=0 to y=a/2 ] and limit of x changes into [ x= y^2/a to x= {a-sqrt(a^2-4y^2) }/2]
Which means {a-sqrt(a^2-4y^2)}/2 is greater than {a+sqrt(a^2-4y^2}/2



In AMC, again we see that {a+sqrt(a^2-4y^2}/2 is taken as lower limit of x.



So, I could not understand how to decide or check which number will be greater and which number will be smaller between {a-sqrt(a^2-4y^2)}/2 is and {a+sqrt(a^2-4y^2}/2



My questions are:-



(i) How can we testify that which number can be taken as upper limit and which number can be taken as lower limit of x in the different terms while changing the order of integration, in accordance with the given question.

(ii) Is there any method to test whether one algebraic expression (or number) is greater than the other in Mathematics? If yes, then please tell me.


P.S.:- Thank You So Much in advance if you willing to amswer my question. Thanks again if you edited my question to make math expressions easily understable, as I don't know how to do it here.



Please Upvote!!😊



Please don't downvote my question, If you find anything wrong in my question or not according to the board, please let me know in the comments. I'll delete the question.



Thank You Everyone for making this board great.










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  • @Joel Pereira Thanks for the edit. It would so humble if you read the whole question and edit it wherever needed. Thanks in advance.
    – Naved THE Sheikh
    2 days ago






  • 2




    I did some of formatting for you. Please read this MathJax reference guide. Using the guide, as well as my edits as some samples, you should be able to improve your own post.
    – zipirovich
    2 days ago












  • Thank You So Much @zipirovich for the above edit and the reference. I'll surely read it and try not to panic other members to format my posts in future. 😊😊😊
    – Naved THE Sheikh
    2 days ago
















0














My math book has a question of changing order of the following integral:



$$int_0^aint_{sqrt{ax-x^2}}^{sqrt{ax}} f(x,y),dx,dy.$$



So basically I have to change $dxdy$ into $dydx$ and their respective limits.



This is an example which is provided in my math textbook related to the problem. (For showing you diagram and make it easier for me to explain my confusion further.)



1]



Image Link



So in my problem, while changing the order of integration, the resultant changed integral order is coming in 3 terms (say, IOnfoot1+IOnfoot2+IOnfoot3). I1 is for OLC, I2 is for AMC and I3 is for BLCM.(Please refer to the diagram, which I shared in the form of Image)



While changing the circle, i.e. $y=sqrt{ax-x^2}$ into form of $x$, we get $x=frac{1}{2}left(apmsqrt{a^2-4y^2}right)$.



I tried to solve it ahead but I stucked on a problem of deciding in ± which to choose to get higher (upper) limit and to get lower limit. But in the answersheet of my Textbook, the following is given:-



In OLC the limit of y changes into [ from y=0 to y=a/2 ] and limit of x changes into [ x= y^2/a to x= {a-sqrt(a^2-4y^2) }/2]
Which means {a-sqrt(a^2-4y^2)}/2 is greater than {a+sqrt(a^2-4y^2}/2



In AMC, again we see that {a+sqrt(a^2-4y^2}/2 is taken as lower limit of x.



So, I could not understand how to decide or check which number will be greater and which number will be smaller between {a-sqrt(a^2-4y^2)}/2 is and {a+sqrt(a^2-4y^2}/2



My questions are:-



(i) How can we testify that which number can be taken as upper limit and which number can be taken as lower limit of x in the different terms while changing the order of integration, in accordance with the given question.

(ii) Is there any method to test whether one algebraic expression (or number) is greater than the other in Mathematics? If yes, then please tell me.


P.S.:- Thank You So Much in advance if you willing to amswer my question. Thanks again if you edited my question to make math expressions easily understable, as I don't know how to do it here.



Please Upvote!!😊



Please don't downvote my question, If you find anything wrong in my question or not according to the board, please let me know in the comments. I'll delete the question.



Thank You Everyone for making this board great.










share|cite|improve this question
























  • @Joel Pereira Thanks for the edit. It would so humble if you read the whole question and edit it wherever needed. Thanks in advance.
    – Naved THE Sheikh
    2 days ago






  • 2




    I did some of formatting for you. Please read this MathJax reference guide. Using the guide, as well as my edits as some samples, you should be able to improve your own post.
    – zipirovich
    2 days ago












  • Thank You So Much @zipirovich for the above edit and the reference. I'll surely read it and try not to panic other members to format my posts in future. 😊😊😊
    – Naved THE Sheikh
    2 days ago














0












0








0







My math book has a question of changing order of the following integral:



$$int_0^aint_{sqrt{ax-x^2}}^{sqrt{ax}} f(x,y),dx,dy.$$



So basically I have to change $dxdy$ into $dydx$ and their respective limits.



This is an example which is provided in my math textbook related to the problem. (For showing you diagram and make it easier for me to explain my confusion further.)



1]



Image Link



So in my problem, while changing the order of integration, the resultant changed integral order is coming in 3 terms (say, IOnfoot1+IOnfoot2+IOnfoot3). I1 is for OLC, I2 is for AMC and I3 is for BLCM.(Please refer to the diagram, which I shared in the form of Image)



While changing the circle, i.e. $y=sqrt{ax-x^2}$ into form of $x$, we get $x=frac{1}{2}left(apmsqrt{a^2-4y^2}right)$.



I tried to solve it ahead but I stucked on a problem of deciding in ± which to choose to get higher (upper) limit and to get lower limit. But in the answersheet of my Textbook, the following is given:-



In OLC the limit of y changes into [ from y=0 to y=a/2 ] and limit of x changes into [ x= y^2/a to x= {a-sqrt(a^2-4y^2) }/2]
Which means {a-sqrt(a^2-4y^2)}/2 is greater than {a+sqrt(a^2-4y^2}/2



In AMC, again we see that {a+sqrt(a^2-4y^2}/2 is taken as lower limit of x.



So, I could not understand how to decide or check which number will be greater and which number will be smaller between {a-sqrt(a^2-4y^2)}/2 is and {a+sqrt(a^2-4y^2}/2



My questions are:-



(i) How can we testify that which number can be taken as upper limit and which number can be taken as lower limit of x in the different terms while changing the order of integration, in accordance with the given question.

(ii) Is there any method to test whether one algebraic expression (or number) is greater than the other in Mathematics? If yes, then please tell me.


P.S.:- Thank You So Much in advance if you willing to amswer my question. Thanks again if you edited my question to make math expressions easily understable, as I don't know how to do it here.



Please Upvote!!😊



Please don't downvote my question, If you find anything wrong in my question or not according to the board, please let me know in the comments. I'll delete the question.



Thank You Everyone for making this board great.










share|cite|improve this question















My math book has a question of changing order of the following integral:



$$int_0^aint_{sqrt{ax-x^2}}^{sqrt{ax}} f(x,y),dx,dy.$$



So basically I have to change $dxdy$ into $dydx$ and their respective limits.



This is an example which is provided in my math textbook related to the problem. (For showing you diagram and make it easier for me to explain my confusion further.)



1]



Image Link



So in my problem, while changing the order of integration, the resultant changed integral order is coming in 3 terms (say, IOnfoot1+IOnfoot2+IOnfoot3). I1 is for OLC, I2 is for AMC and I3 is for BLCM.(Please refer to the diagram, which I shared in the form of Image)



While changing the circle, i.e. $y=sqrt{ax-x^2}$ into form of $x$, we get $x=frac{1}{2}left(apmsqrt{a^2-4y^2}right)$.



I tried to solve it ahead but I stucked on a problem of deciding in ± which to choose to get higher (upper) limit and to get lower limit. But in the answersheet of my Textbook, the following is given:-



In OLC the limit of y changes into [ from y=0 to y=a/2 ] and limit of x changes into [ x= y^2/a to x= {a-sqrt(a^2-4y^2) }/2]
Which means {a-sqrt(a^2-4y^2)}/2 is greater than {a+sqrt(a^2-4y^2}/2



In AMC, again we see that {a+sqrt(a^2-4y^2}/2 is taken as lower limit of x.



So, I could not understand how to decide or check which number will be greater and which number will be smaller between {a-sqrt(a^2-4y^2)}/2 is and {a+sqrt(a^2-4y^2}/2



My questions are:-



(i) How can we testify that which number can be taken as upper limit and which number can be taken as lower limit of x in the different terms while changing the order of integration, in accordance with the given question.

(ii) Is there any method to test whether one algebraic expression (or number) is greater than the other in Mathematics? If yes, then please tell me.


P.S.:- Thank You So Much in advance if you willing to amswer my question. Thanks again if you edited my question to make math expressions easily understable, as I don't know how to do it here.



Please Upvote!!😊



Please don't downvote my question, If you find anything wrong in my question or not according to the board, please let me know in the comments. I'll delete the question.



Thank You Everyone for making this board great.







integration definite-integrals improper-integrals indefinite-integrals






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edited 2 days ago









zipirovich

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asked 2 days ago









Naved THE Sheikh

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  • @Joel Pereira Thanks for the edit. It would so humble if you read the whole question and edit it wherever needed. Thanks in advance.
    – Naved THE Sheikh
    2 days ago






  • 2




    I did some of formatting for you. Please read this MathJax reference guide. Using the guide, as well as my edits as some samples, you should be able to improve your own post.
    – zipirovich
    2 days ago












  • Thank You So Much @zipirovich for the above edit and the reference. I'll surely read it and try not to panic other members to format my posts in future. 😊😊😊
    – Naved THE Sheikh
    2 days ago


















  • @Joel Pereira Thanks for the edit. It would so humble if you read the whole question and edit it wherever needed. Thanks in advance.
    – Naved THE Sheikh
    2 days ago






  • 2




    I did some of formatting for you. Please read this MathJax reference guide. Using the guide, as well as my edits as some samples, you should be able to improve your own post.
    – zipirovich
    2 days ago












  • Thank You So Much @zipirovich for the above edit and the reference. I'll surely read it and try not to panic other members to format my posts in future. 😊😊😊
    – Naved THE Sheikh
    2 days ago
















@Joel Pereira Thanks for the edit. It would so humble if you read the whole question and edit it wherever needed. Thanks in advance.
– Naved THE Sheikh
2 days ago




@Joel Pereira Thanks for the edit. It would so humble if you read the whole question and edit it wherever needed. Thanks in advance.
– Naved THE Sheikh
2 days ago




2




2




I did some of formatting for you. Please read this MathJax reference guide. Using the guide, as well as my edits as some samples, you should be able to improve your own post.
– zipirovich
2 days ago






I did some of formatting for you. Please read this MathJax reference guide. Using the guide, as well as my edits as some samples, you should be able to improve your own post.
– zipirovich
2 days ago














Thank You So Much @zipirovich for the above edit and the reference. I'll surely read it and try not to panic other members to format my posts in future. 😊😊😊
– Naved THE Sheikh
2 days ago




Thank You So Much @zipirovich for the above edit and the reference. I'll surely read it and try not to panic other members to format my posts in future. 😊😊😊
– Naved THE Sheikh
2 days ago










1 Answer
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The question in the textbook is stated incorrectly. The original order of integration is $dydx$ NOT $dxdy$. This is because we are integrating $y$ first on the inner integral and then $x$. It cannot be $dxdy$ because we cannot integrate with respect to $x$ and have the limits of the integral be written in terms of $x$.



i) Typically we integrate on the $x$ axis from left to right. By doing so we can see that the graph $y =sqrt{2ax} $ is the leftmost curve. Therefore the lower limit will be $x=frac{y^2} {2a} $. Similarly, moving rightwards along the $x$ axis we 'hit' the left section of the circular region next. For the curve $(x-a)^2 + y^2 = a^2$ we can rearrange to get $x = a{+/-} {sqrt{a^2-y^2}} $. This curve takes the positive sqrt for the right half of the circle and the negative sqrt for the left half of the circle. As we need the equation of the left half of the circle, we need to take $x = a-sqrt{a^2-y^2} $ for the upper limit. This does not imply that this value is greater than the other.



ii) In general taking the positive sqrt will always result in a greater number than the negative sqrt, but when integrating, we do not always want the greater value, we want the equation of the curve we are using to integrate.






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    The question in the textbook is stated incorrectly. The original order of integration is $dydx$ NOT $dxdy$. This is because we are integrating $y$ first on the inner integral and then $x$. It cannot be $dxdy$ because we cannot integrate with respect to $x$ and have the limits of the integral be written in terms of $x$.



    i) Typically we integrate on the $x$ axis from left to right. By doing so we can see that the graph $y =sqrt{2ax} $ is the leftmost curve. Therefore the lower limit will be $x=frac{y^2} {2a} $. Similarly, moving rightwards along the $x$ axis we 'hit' the left section of the circular region next. For the curve $(x-a)^2 + y^2 = a^2$ we can rearrange to get $x = a{+/-} {sqrt{a^2-y^2}} $. This curve takes the positive sqrt for the right half of the circle and the negative sqrt for the left half of the circle. As we need the equation of the left half of the circle, we need to take $x = a-sqrt{a^2-y^2} $ for the upper limit. This does not imply that this value is greater than the other.



    ii) In general taking the positive sqrt will always result in a greater number than the negative sqrt, but when integrating, we do not always want the greater value, we want the equation of the curve we are using to integrate.






    share|cite|improve this answer










    New contributor




    Peter Foreman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      4














      The question in the textbook is stated incorrectly. The original order of integration is $dydx$ NOT $dxdy$. This is because we are integrating $y$ first on the inner integral and then $x$. It cannot be $dxdy$ because we cannot integrate with respect to $x$ and have the limits of the integral be written in terms of $x$.



      i) Typically we integrate on the $x$ axis from left to right. By doing so we can see that the graph $y =sqrt{2ax} $ is the leftmost curve. Therefore the lower limit will be $x=frac{y^2} {2a} $. Similarly, moving rightwards along the $x$ axis we 'hit' the left section of the circular region next. For the curve $(x-a)^2 + y^2 = a^2$ we can rearrange to get $x = a{+/-} {sqrt{a^2-y^2}} $. This curve takes the positive sqrt for the right half of the circle and the negative sqrt for the left half of the circle. As we need the equation of the left half of the circle, we need to take $x = a-sqrt{a^2-y^2} $ for the upper limit. This does not imply that this value is greater than the other.



      ii) In general taking the positive sqrt will always result in a greater number than the negative sqrt, but when integrating, we do not always want the greater value, we want the equation of the curve we are using to integrate.






      share|cite|improve this answer










      New contributor




      Peter Foreman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





















        4












        4








        4






        The question in the textbook is stated incorrectly. The original order of integration is $dydx$ NOT $dxdy$. This is because we are integrating $y$ first on the inner integral and then $x$. It cannot be $dxdy$ because we cannot integrate with respect to $x$ and have the limits of the integral be written in terms of $x$.



        i) Typically we integrate on the $x$ axis from left to right. By doing so we can see that the graph $y =sqrt{2ax} $ is the leftmost curve. Therefore the lower limit will be $x=frac{y^2} {2a} $. Similarly, moving rightwards along the $x$ axis we 'hit' the left section of the circular region next. For the curve $(x-a)^2 + y^2 = a^2$ we can rearrange to get $x = a{+/-} {sqrt{a^2-y^2}} $. This curve takes the positive sqrt for the right half of the circle and the negative sqrt for the left half of the circle. As we need the equation of the left half of the circle, we need to take $x = a-sqrt{a^2-y^2} $ for the upper limit. This does not imply that this value is greater than the other.



        ii) In general taking the positive sqrt will always result in a greater number than the negative sqrt, but when integrating, we do not always want the greater value, we want the equation of the curve we are using to integrate.






        share|cite|improve this answer










        New contributor




        Peter Foreman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        The question in the textbook is stated incorrectly. The original order of integration is $dydx$ NOT $dxdy$. This is because we are integrating $y$ first on the inner integral and then $x$. It cannot be $dxdy$ because we cannot integrate with respect to $x$ and have the limits of the integral be written in terms of $x$.



        i) Typically we integrate on the $x$ axis from left to right. By doing so we can see that the graph $y =sqrt{2ax} $ is the leftmost curve. Therefore the lower limit will be $x=frac{y^2} {2a} $. Similarly, moving rightwards along the $x$ axis we 'hit' the left section of the circular region next. For the curve $(x-a)^2 + y^2 = a^2$ we can rearrange to get $x = a{+/-} {sqrt{a^2-y^2}} $. This curve takes the positive sqrt for the right half of the circle and the negative sqrt for the left half of the circle. As we need the equation of the left half of the circle, we need to take $x = a-sqrt{a^2-y^2} $ for the upper limit. This does not imply that this value is greater than the other.



        ii) In general taking the positive sqrt will always result in a greater number than the negative sqrt, but when integrating, we do not always want the greater value, we want the equation of the curve we are using to integrate.







        share|cite|improve this answer










        New contributor




        Peter Foreman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 days ago









        jayant98

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        answered 2 days ago









        Peter Foreman

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        Peter Foreman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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        Peter Foreman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.






























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