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A
0
=
α
−
1
π
∫
0
π
d
Z
¯
d
x
(
θ
)
d
θ
{\displaystyle A_{0}=\alpha -{\frac {1}{\pi }}\int _{0}^{\pi }{\frac {d{\overline {Z}}}{dx}}(\theta )d\theta }
A
n
=
2
π
∫
0
π
d
Z
¯
d
x
(
θ
)
cos
(
n
θ
)
d
θ
{\displaystyle 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
)
=
∫
0
x
w
(
x
∗
)
d
x
∗
=
∫
0
x
c
1
−
4
x
∗
2
b
2
d
x
∗
=
1
/
2
c
x
1
−
4
x
2
b
2
+
1
/
4
c
arctan
(
2
b
−
2
x
1
1
−
4
x
2
b
2
)
1
b
−
2
{\displaystyle 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
)
{\displaystyle L(x)}
c
=
2
w
e
i
g
h
t
n
m
a
x
a
π
{\displaystyle c={\frac {2\,\mathrm {weight} \,n_{max}}{a\pi }}}
b
{\displaystyle b}
x
∗
{\displaystyle x^{*}}
τ
{\displaystyle \tau }
c
e
n
t
r
o
i
d
=
∫
0
x
x
∗
c
1
−
4
x
∗
2
b
2
d
x
∗
∫
0
x
c
1
−
4
x
∗
2
b
2
d
x
∗
=
1
/
12
(
2
x
−
b
)
(
2
x
+
b
)
c
−
−
b
2
+
4
x
2
b
2
(
1
/
2
c
x
1
−
4
x
2
b
2
+
1
/
4
c
arctan
(
2
b
−
2
x
1
1
−
4
x
2
b
2
)
1
b
−
2
)
−
1
{\displaystyle \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}}
is the total force at point x
is the span
is the integration variable (kinda like how sometimes you integrate in 
from 0 to t)