When heuristics go bad | Colin

Last week, I wrote about the volume and outer surface area of a spherical cap using different methods, both of which gave the volume as $V = \frac{\pi}{3}R^3 (1-\cos(\alpha))^2(2-\cos(\alpha))$ and the surface area as $A_o = 2\pi R^2 (1-\cos(\alpha))$.

All very nice; however, one of my most beloved heuristics fails in this case.

If you differentiate the area of a circle ($\pi r^2$) with respect to the radius, you get $2\pi r$ – the circumference. If you differentiate the volume of a sphere ($\frac{4}{3}\pi r^3$), you get $4\pi r^2$, the surface area. In a way that can be nicely extended…

Continue reading at:


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s