User:!jim/Sandbox

$$A_0=\alpha-\frac{1}{\pi}\int_0^\pi\frac{d\overline{Z}}{dx}(\theta)d\theta$$

$$A_n=\frac{2}{\pi}\int_0^\pi\frac{d\overline{Z}}{dx}(\theta)\cos(n\theta) d\theta$$

Load at point x
$$ L(x)=\int_0^x w(x^*) dx^*=\int_0^x c\sqrt{1-\frac{4x^{*2}}{b^2}} dx^*=1/2\,cx\sqrt {1-4\,{\frac {{x}^{2}}{{b}^{2}}}}+1/4\,c\arctan \left( 2 \,\sqrt {{b}^{-2}}x{\frac {1}{\sqrt {1-4\,{\frac {{x}^{2}}{{b}^{2}}}}} } \right) {\frac {1}{\sqrt {{b}^{-2}}}}

$$

where:
 * $$L(x)$$ is the total force at point x
 * $$c=\frac{2\,\mathrm{weight}\, n_{max}}{a\pi}$$
 * $$b$$ is the span
 * $$x^*$$ is the integration variable (kinda like how sometimes you integrate in $$\tau$$ from 0 to t)

$$ \mathrm{centroid}=\frac{\int_0^x x^*c\sqrt{1-\frac{4x^{*2}}{b^2}} dx^*}{\int_0^x c\sqrt{1-\frac{4x^{*2}}{b^2}} dx^*}=1/12\, \left( 2\,x-b \right) \left( 2\,x+b \right) c\sqrt {-{\frac {- {b}^{2}+4\,{x}^{2}}{{b}^{2}}}} \left( 1/2\,cx\sqrt {1-4\,{\frac {{x}^{ 2}}{{b}^{2}}}}+1/4\,c\arctan \left( 2\,\sqrt {{b}^{-2}}x{\frac {1}{ \sqrt {1-4\,{\frac {{x}^{2}}{{b}^{2}}}}}} \right) {\frac {1}{\sqrt {{b }^{-2}}}} \right) ^{-1}

$$