Integration by substitution to find the arc length of an ellipse in polar form.











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I have that $l/r = 1+e.cos(x)$, for $l = a(1-e^2)$ (constant).
The question asks for the mean distance over angle of the planet from the sun, where the planet moves on an elliptical orbit with the sun at a focus. and gives the formula (and answer):
$$frac{1}{2pi} int^{2pi}_0 r , dx = a(1-e^2)^{frac{1}{2}}$$



I know I am meant to use the substitution $t = tan(x/2)$, and then use a substitution again later, but my integral results in a function arctan, and I don't know how to get a constant from that.



Any help much appreciated, thank you.






calculus conic-sections polar-coordinates arc-length







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  • 1




    Can you please clarify? Total distance of what? That's not the correct expression for the arclength of an ellipse, which involves elliptic integrals; see here, for example: math.stackexchange.com/questions/433094/…
    – Hans Lundmark
    11 hours ago










  • Sorry - the question I have (undergraduate Physics) this (I missed a 2pi):
    – AMath
    10 hours ago










  • int^{2pi}_0 r dx = 2pi a(1-e^2)^{frac{1}{2}}
    – AMath
    10 hours ago










  • Where x is the angle. Sorry - can't work out how to add a picture, hence the latex type!
    – AMath
    10 hours ago






  • 1




    Oh! The original question asks for the mean distance over angle of the planet from the sun and gives the formula (and answer):
    – AMath
    8 hours ago















up vote
0
down vote

favorite












I have that $l/r = 1+e.cos(x)$, for $l = a(1-e^2)$ (constant).
The question asks for the mean distance over angle of the planet from the sun, where the planet moves on an elliptical orbit with the sun at a focus. and gives the formula (and answer):
$$frac{1}{2pi} int^{2pi}_0 r , dx = a(1-e^2)^{frac{1}{2}}$$



I know I am meant to use the substitution $t = tan(x/2)$, and then use a substitution again later, but my integral results in a function arctan, and I don't know how to get a constant from that.



Any help much appreciated, thank you.






calculus conic-sections polar-coordinates arc-length







share|cite|improve this question




















  • 1




    Can you please clarify? Total distance of what? That's not the correct expression for the arclength of an ellipse, which involves elliptic integrals; see here, for example: math.stackexchange.com/questions/433094/…
    – Hans Lundmark
    11 hours ago










  • Sorry - the question I have (undergraduate Physics) this (I missed a 2pi):
    – AMath
    10 hours ago










  • int^{2pi}_0 r dx = 2pi a(1-e^2)^{frac{1}{2}}
    – AMath
    10 hours ago










  • Where x is the angle. Sorry - can't work out how to add a picture, hence the latex type!
    – AMath
    10 hours ago






  • 1




    Oh! The original question asks for the mean distance over angle of the planet from the sun and gives the formula (and answer):
    – AMath
    8 hours ago













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have that $l/r = 1+e.cos(x)$, for $l = a(1-e^2)$ (constant).
The question asks for the mean distance over angle of the planet from the sun, where the planet moves on an elliptical orbit with the sun at a focus. and gives the formula (and answer):
$$frac{1}{2pi} int^{2pi}_0 r , dx = a(1-e^2)^{frac{1}{2}}$$



I know I am meant to use the substitution $t = tan(x/2)$, and then use a substitution again later, but my integral results in a function arctan, and I don't know how to get a constant from that.



Any help much appreciated, thank you.






calculus conic-sections polar-coordinates arc-length







share|cite|improve this question















I have that $l/r = 1+e.cos(x)$, for $l = a(1-e^2)$ (constant).
The question asks for the mean distance over angle of the planet from the sun, where the planet moves on an elliptical orbit with the sun at a focus. and gives the formula (and answer):
$$frac{1}{2pi} int^{2pi}_0 r , dx = a(1-e^2)^{frac{1}{2}}$$



I know I am meant to use the substitution $t = tan(x/2)$, and then use a substitution again later, but my integral results in a function arctan, and I don't know how to get a constant from that.



Any help much appreciated, thank you.






calculus conic-sections polar-coordinates arc-length




calculus conic-sections polar-coordinates arc-length






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 8 hours ago









Kevin

5,365822




5,365822










asked 11 hours ago









AMath

13




13








  • 1




    Can you please clarify? Total distance of what? That's not the correct expression for the arclength of an ellipse, which involves elliptic integrals; see here, for example: math.stackexchange.com/questions/433094/…
    – Hans Lundmark
    11 hours ago










  • Sorry - the question I have (undergraduate Physics) this (I missed a 2pi):
    – AMath
    10 hours ago










  • int^{2pi}_0 r dx = 2pi a(1-e^2)^{frac{1}{2}}
    – AMath
    10 hours ago










  • Where x is the angle. Sorry - can't work out how to add a picture, hence the latex type!
    – AMath
    10 hours ago






  • 1




    Oh! The original question asks for the mean distance over angle of the planet from the sun and gives the formula (and answer):
    – AMath
    8 hours ago














  • 1




    Can you please clarify? Total distance of what? That's not the correct expression for the arclength of an ellipse, which involves elliptic integrals; see here, for example: math.stackexchange.com/questions/433094/…
    – Hans Lundmark
    11 hours ago










  • Sorry - the question I have (undergraduate Physics) this (I missed a 2pi):
    – AMath
    10 hours ago










  • int^{2pi}_0 r dx = 2pi a(1-e^2)^{frac{1}{2}}
    – AMath
    10 hours ago










  • Where x is the angle. Sorry - can't work out how to add a picture, hence the latex type!
    – AMath
    10 hours ago






  • 1




    Oh! The original question asks for the mean distance over angle of the planet from the sun and gives the formula (and answer):
    – AMath
    8 hours ago








1




1




Can you please clarify? Total distance of what? That's not the correct expression for the arclength of an ellipse, which involves elliptic integrals; see here, for example: math.stackexchange.com/questions/433094/…
– Hans Lundmark
11 hours ago




Can you please clarify? Total distance of what? That's not the correct expression for the arclength of an ellipse, which involves elliptic integrals; see here, for example: math.stackexchange.com/questions/433094/…
– Hans Lundmark
11 hours ago












Sorry - the question I have (undergraduate Physics) this (I missed a 2pi):
– AMath
10 hours ago




Sorry - the question I have (undergraduate Physics) this (I missed a 2pi):
– AMath
10 hours ago












int^{2pi}_0 r dx = 2pi a(1-e^2)^{frac{1}{2}}
– AMath
10 hours ago




int^{2pi}_0 r dx = 2pi a(1-e^2)^{frac{1}{2}}
– AMath
10 hours ago












Where x is the angle. Sorry - can't work out how to add a picture, hence the latex type!
– AMath
10 hours ago




Where x is the angle. Sorry - can't work out how to add a picture, hence the latex type!
– AMath
10 hours ago




1




1




Oh! The original question asks for the mean distance over angle of the planet from the sun and gives the formula (and answer):
– AMath
8 hours ago




Oh! The original question asks for the mean distance over angle of the planet from the sun and gives the formula (and answer):
– AMath
8 hours ago















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