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Questions and Answers
The volume of a cone can be derived by integrating the area of circular ______ perpendicular to the height of the cone.
cross-sections
The volume of each disk is approximately ______x² dy.
Ï€
The radius x of each disk is proportional to the distance ______ from the apex of the cone.
y
Integrate with respect to ______ from 0 to h.
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The volume of a cone is: V = (1/3)______r²h where r is the radius of the base and h is the height of the cone.
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Study Notes
Volume of a Cone: Formula Derivation
Introduction
The volume of a cone can be derived by integrating the area of circular cross-sections perpendicular to the height of the cone.
Derivation
- Consider a cone with radius
rand heighth. - Divide the cone into thin circular disks of thickness
dyand radiusx. - The volume of each disk is approximately
πx² dy. - The radius
xof each disk is proportional to the distanceyfrom the apex of the cone:x = (r/h) \* y. - Substitute the expression for
xinto the volume formula:π((r/h) \* y)² dy. - Integrate with respect to
yfrom0toh: ∫[0,h]π((r/h) \* y)² dy = (1/3)πr²h.
Formula
The volume of a cone is: V = (1/3)πr²h
where r is the radius of the base and h is the height of the cone.
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Description
Derive the formula for the volume of a cone by integrating the area of circular cross-sections. Learn how to calculate the volume of a cone with radius and height.
