(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!)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**