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
edited 8 hours ago
Kevin
5,365822
asked 11 hours ago
AMath
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show 5 more comments
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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
edited 8 hours ago
Kevin
5,365822
asked 11 hours ago
AMath
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
|
show 5 more comments
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
edited 8 hours ago
Kevin
5,365822
asked 11 hours ago
AMath
13
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
edited 8 hours ago
Kevin
5,365822
asked 11 hours ago
AMath
13
edited 8 hours ago
Kevin
5,365822
asked 11 hours ago
AMath
13
edited 8 hours ago
Kevin
5,365822
edited 8 hours ago
Kevin
5,365822
edited 8 hours ago
Kevin
5,365822
5,365822
asked 11 hours ago
AMath
13
asked 11 hours ago
AMath
13
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
|
show 5 more comments
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
|
show 5 more comments
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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