Surface area formula proof - integrate circumference of volume of revolution

In summary, the conversation discusses the attempt to solve for the surface area of a cone using the formula πrl and the concept of volume of revolution. The attempt involved integration and resulted in the incorrect formula of πrh. The question remains as to why this approach did not work in obtaining the correct formula.
  • #1
whatisreality
290
1

Homework Statement


I believe this is intended to be a proof of the formula πrl, surface area of a cone.


Homework Equations


A complete volume of revolution gives you a cone - the height h is the x value on a graph, the radius r is the y value. The y intercept is zero, therefore y=r/hx . All integration is with lim h→0


The Attempt at a Solution


When I tried this, I got:
∫2πydx
=2π∫r/hx dx
=2πr/h∫x dx
=2πr/h ∫x2/2
=πrh! So why has this not worked? I need this to give πrl!
 
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  • #2
No-one?
 

Related to Surface area formula proof - integrate circumference of volume of revolution

What is the surface area formula for a volume of revolution?

The surface area formula for a volume of revolution is given by the integral of the circumference of the cross sections of the shape as it is rotated around an axis.

What is the significance of integrating the circumference in the surface area formula?

Integrating the circumference allows us to take into account the changing shape and size of the cross sections as the shape is rotated, resulting in a more accurate calculation of the surface area.

Can the surface area formula be used for any shape?

Yes, the surface area formula can be used for any shape that can be rotated around an axis, as long as the cross sections are circular in shape.

How is the surface area formula derived?

The surface area formula is derived using calculus and the concept of integration, by breaking down the shape into infinitely small cross sections and summing their surface areas.

Are there any limitations to using the surface area formula?

One limitation is that it can only be used for shapes with circular cross sections. Additionally, it may be difficult to use for shapes with irregular or complex cross sections.

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