How to calculates the volume of an ellipsoidal cap
How to calculates the volume of an ellipsoidal cap
Z
c V(z) = S(z) dz
c-h
hc xy z
S(z) = xy Y
equation of an ellipsoid
equation of an ellipse along the X-axis
y=0
equation of an ellipse along the Y-axis
x=0
b
X
a
area of a vertical ellipse to the Z-axis
volume of an ellipsoidal cap
volume of an ellipsoid
2015. 10. 29. keisan
x2 y2 z2 a2 + b2 + c2 = 1
x2 z2 a2 + c2 = 1 y2 z2 b2 + c2 = 1
a
x = c
c2-z2
b y= c
c2-z2
S(z) = xy z2
= ab 1- c2
c
c
V(z) = S(z)dz = ab
c-h
c-h
z2 1- c2
dz
c3
(c-h)3
= ab c- 3c2 -(c-h)+ 3c2
abh2 = 3c2 ( 3c-h)
c
c
V(z) = S(z)dz = ab
-c
-c
z2 1- c2
dz
4abc = 3
................
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