Integrating an Ellipse: Calculating the Width of a Melding Pillar and Roof

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The discussion focuses on calculating the integral of an ellipse to estimate the width of a melding pillar and roof. The user is trying to integrate the equation involving elliptic integrals, which cannot be expressed in terms of elementary functions. They express confusion over substitution methods and whether to include differential elements in their calculations. The conversation highlights that MATLAB has a built-in function, 'ellipke', for elliptic integrals, while standard integrals for circles can be solved through substitution. The user aims to derive a formula for the width of the pillar at the top, considering the dimensions of the ellipse.
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Hi

Im trying to estimate a semicircle or ellips of a kind with an integral.
And right now I'm trying to get the integral of a ellips.

I need to integrate the equation


x(\theta) = \int_0^{\pi/2} \frac {d\theta}{a\sqrt{1-sin^2(\theta)/b^2}}

I tried http://integrals.wolfram.com/index.jsp?expr=1/sqrt(1-sin^2(x)/b^2)&random=false" but i don't understand the result or how to integrate the result.

Thanks for any help
 
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That's one of the reasons elliptic integrals are called <special functions>, because they can't be written in terms of 'elementary' functions, such as polynomials, sin, cos, e^x, ln, sinh, cosh, ...The perimeter of an ellipse is an elliptic integral.
 
so there is no way for me to just but in the boundaries? i want to implement this in a MATLAB function, later on does it mean that i have to use a nummerical method to solve it then?

and what do you do if you if you have a circle is that also impossible to solve? because the only difference is the a and b constants.
 
The special function you want is already implemented in Matlab. It's called ellipke in Matlab.
 
The circle is ok. It's a standard integral solvable by a substitution.
 
I think i might have made a mistake when substituting.

I have
<br /> <br /> \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1<br /> <br />

And i substitute by

<br /> y = sin(\theta)<br /> x = cos(\theta)<br />


So when i do the substitution should i include that dy = cos(\theta)d\theta or can i just substitute
<br /> x = \sqrt{a^2 + \frac{a^2}{b^2}y^2} → \sqrt{a^2 + \frac{a^2}{b^2}sin(\theta)^2}<br />

and then integrate as x = \int_0^b\frac{dy}{f(y)} or am i doing some big mistakes?
 
I think i might have made a mistake when substituting.

I have
<br /> <br /> \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1<br /> <br />

And i substitute by

<br /> y = sin(\theta)<br /> x = cos(\theta)<br />


So when i do the substitution should i include that dy = cos(\theta)d\theta or can i just substitute
<br /> x = \sqrt{a^2 + \frac{a^2}{b^2}y^2} → \sqrt{a^2 + \frac{a^2}{b^2}sin(\theta)^2}<br />

and then integrate as g = \int_0^b\frac{dy}{c-f(y)}

where c is a fixed length and f(y) is the function x. or am i doing some big mistakes?


What i want to do is calculate the width of a pillar at the top as it is melding together with the roof. I assume that the arc is in the form of an ellipse so i have

c = the length of the pillar at the top.
b = the length between when the pillar starts curving to the roof
a = c - half of the pillar length

g = c - x(y)

and then integrate as 1/g.

and then multiple by to get the other side.
 
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