What is the Correct Formula for the Circumference of an Ellipse?

In summary: So the integral should be 1 - (sin(θ))^2 + (b^2/a^2)(sin(θ))^2.No! It should be 1 - (sin(θ))^2 + (b^2/a^2)(sin(θ))^2! You are subtracting k. That's just being sloppy.
  • #1
emc92
33
0
all of my work so far is in the picture. I'm stuck on what i should do next.
 

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  • #2
your handwriting is incredibly neat, good show!
 
  • #3
You are making the substitution [itex]x= a sin\theta[/itex] but then your integral has both x and [itex]\theta[/itex]. That's not right.

However, I would advise using the parametric equations [itex]x= a sin(\theta)[/itex], [itex]y= b cos(\theta)[/itex] rather than that complicated equation.
 
  • #4
HallsofIvy said:
You are making the substitution [itex]x= a sin\theta[/itex] but then your integral has both x and [itex]\theta[/itex]. That's not right.

However, I would advise using the parametric equations [itex]x= a sin(\theta)[/itex], [itex]y= b cos(\theta)[/itex] rather than that complicated equation.

I'm not sure I understand what you mean by the parametric equations.. how does that fit into what I already have?
 
  • #5
genericusrnme said:
your handwriting is incredibly neat, good show!

thanks!
 
  • #6
emc92 said:
I'm not sure I understand what you mean by the parametric equations.. how does that fit into what I already have?

You were doing alright until after you drew the triangle. But you want to get rid of all of the x's in the thing you are integrating. And you never used dx=a cos(θ) dθ, probably because you weren't writing the dx in the integration. You need that.
 
  • #7
Dick said:
You were doing alright until after you drew the triangle. But you want to get rid of all of the x's in the thing you are integrating. And you never used dx=a cos(θ) dθ, probably because you weren't writing the dx in the integration. You need that.

oh darn! i did forget. okie well, now that I have dθ in there and i changed x^2, i still have a really ugly equation.. what should i do next?
 

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  • #8
emc92 said:
oh darn! i did forget. okie well, now that I have dθ in there and i changed x^2, i still have a really ugly equation.. what should i do next?

Bring the cos(θ) inside the square root where it becomes cos^2(θ). And i) replacing the a^2 with x^2/sin^2(θ) doesn't do you any good and ii) somewhere you missed cancelling an a^2.
 
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  • #9
i completely reworked it, and it looks so much better now! lol.
now i have integral from 0 to a of sqrt(1- (b^2/a^2)cos(θ))
 
  • #10
emc92 said:
i completely reworked it, and it looks so much better now! lol.
now i have integral from 0 to a of sqrt(1- (b^2/a^2)cos(θ))

I thought you were supposed to get the integral of sqrt(1-k*sin^2(θ))??
 
  • #11
right. k = 1-(b^2/a^2). did i cancel sin^2(θ) instead of cos(θ)?
 
  • #12
emc92 said:
right. k = 1-(b^2/a^2). did i cancel sin^2(θ) instead of cos(θ)?

Hard to say. What did you do?
 
  • #13
Dick said:
Hard to say. What did you do?

i've completely messed up. i don't know what to use for substitutions.. i always end up in the same place. and i don't know where dx = a cos(θ) dθ fits into all of this.

sorry I'm a lot confused!
 

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  • #14
emc92 said:
i've completely messed up. i don't know what to use for substitutions.. i always end up in the same place. and i don't know where dx = a cos(θ) dθ fits into all of this.

sorry I'm a lot confused!

You are going in circles. Look you've got [itex]\int \sqrt{1+\frac{b^2 \sin^2{\theta}}{a^2 \cos^2{\theta}}} a \cos{\theta} d\theta[/itex]. Bring the cos into the square root, so you've got [itex]\int \sqrt{(1+\frac{b^2 \sin^2{\theta}}{a^2 \cos^2{\theta}}) \cos^2{\theta}} a d\theta[/itex]. Simplify inside the radical. Then use your trig identity and change the x limits to theta limits.
 
  • #15
but if k = 1 - (b^2/a^2) and under the radical says 1-ksin^2(θ), shouldn't the end result under the radical, when expanded, be 1 - (sin(θ))^2- (b^2/a^2)(sin(θ))^2?

and where does the 4 outside the integral come from?
 
  • #16
emc92 said:
but if k = 1 - (b^2/a^2) and under the radical says 1-ksin^2(θ), shouldn't the end result under the radical, when expanded, be 1 - (sin(θ))^2- (b^2/a^2)(sin(θ))^2?

and where does the 4 outside the integral come from?

No! It should be 1 - (sin(θ))^2 + (b^2/a^2)(sin(θ))^2! You are subtracting k. That's just being sloppy. And if you are integrating x from 0 to a you are only integrating over 1/4 of the ellipse. You are just covering the first quadrant.
 

1. What is the formula for calculating the circumference of an ellipse?

The formula for calculating the circumference of an ellipse is 2π√((a^2+b^2)/2), where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

2. How is the circumference of an ellipse different from a circle?

The circumference of an ellipse is the distance around the outside of the ellipse, while the circumference of a circle is the distance around the outside of a perfect round shape. The circumference of an ellipse is also longer than that of a circle with the same diameter.

3. Can the circumference of an ellipse be larger than the perimeter of its circumscribing rectangle?

Yes, the circumference of an ellipse can be larger than the perimeter of its circumscribing rectangle. This is because the ellipse's shape is not confined to a rectangle, and its curved sides can extend beyond the straight edges of the rectangle.

4. How do you find the circumference of an ellipse when only the length of its major and minor axes are given?

If only the length of the major and minor axes are given, the circumference of an ellipse can be approximated using the formula π(3(a+b)-√((3a+b)(a+3b))), where a is the length of the major axis and b is the length of the minor axis.

5. Why is the circumference of an ellipse important in real-world applications?

The circumference of an ellipse is important in real-world applications because it can be used to calculate the perimeter of many natural and man-made objects, such as the orbit of planets and the shape of satellite dishes. It also helps in determining the length of fencing needed to enclose an elliptical area or the amount of material needed to create an elliptical structure.

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