(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a vertical cone of height h whose horizontal cross-section is an ellipse and whose base is the ellipse with major and minor semi-axes α and β. Verify that the volume of the cone is παβh/3.

[ Hint: The area of an ellipse with major and minor semi-axes α and β is παβ. ]

2. Relevant equations

V = ∫A(y) dy (from c to d)

V = ∫π(radius)² dy (from c to d)

3. The attempt at a solution

It says that the cone is upright, so I'm assuming it wants the cone rotated about the y-axis.

V = ∫A(y) dy

V = ∫π(radius)² dy

Using similar triangles:

x/y = r/h

x = ry/h

V = π∫(ry/h)² dy (the integral is now from 0 to h (c = 0, d = h))

V = π∫(r²y²/h²) dy

V = (πr²/h²)∫(y²) dy (since pi, r, and h are all constants)

At this point I'm not sure where to go. Do I take the integral of y²? How do I incorporate α and β into this integral? (As a side note, I'm very new to these forums and if I've done anything wrong I apologize. I'm not used to writing out integrals on the computer and if the notation is not optimal I'm sorry!)

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# Solid of revolution question: verify that the volume of the cone is παβh/3

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