(adsbygoogle = window.adsbygoogle || []).push({}); Surface area of a cone--inconsistency?

Geometry tells us that the surface area of a cone with a circular base is

[tex]SA = \pi rs[/tex]

where s is the slant height of the cone, or

[tex]SA = \pi r \sqrt{r^2 + h^2}[/tex]

Take a cone with a circular base of radius 1 and a height of 4. This formula tells us that

[tex]SA = \pi\sqrt{17}[/tex]

However, trying to aproach this problem from a calculus standpoint:

[tex]SA = \int 2 \pi r dh[/tex]

Considering the cone as a triangle for a moment, we have the point (0,4) on the h-axis and (1,0) and the r-axis (where h is the analogue of y and r the analogue of x), and a line connecting them.

This leads to a linear relationship between r and h, such that:

[tex]h = -4r+4[/tex]

or

[tex]r = (0.25)(4-h)[/tex]

Integrating:

[tex]2\pi\int^{4}_{0}(0.25)(4-h)dh[/tex]

=[tex]0.5\pi(4h - 0.5h^2)[/tex]

evaluated from h=0 to 4. However, this yields [tex]4\pi[/tex], not [tex]\pi\sqrt{17}[/tex].

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# Surface area of a cone-inconsistency?

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