# Integration of an ellips

• Mamed

#### Mamed

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
$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$

And i substitute by

$y = sin(\theta) x = cos(\theta)$

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

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
$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$

And i substitute by

$y = sin(\theta) x = cos(\theta)$

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

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.