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mclame22
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Homework Statement
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. ]
Homework Equations
V = ∫A(y) dy (from c to d)
V = ∫π(radius)² dy (from c to d)
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!)