How do you calculate the volume of a ellipsoidal dome?

Answer 1

The answer depends on what you know about the shape of the ellipsoid.

Suppose you have a horizontal x-axis and a vertical y-axis.

Also suppose the equation of a vertical cross section of the ellipsoid is (x/p)2 + (y/q)2 = r2

You need to solve this equation for y at the apex (y = a) and base (y = b) of the dome.

Next, express x2 in terms of y:

x2 = p2*[r2 - (y/q)2]

Finally, the volume is the integral of pi*x2 = pi* p2*[r2 - (y/q)2], with respect to y, between the limits y = a and y = b.