How to calculates the volume of an ellipsoidal cap

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´╗┐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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