The Unbelievably Easy Way To Make The Most Out Of Your Travels
We all love to travel, but how much time can we afford to divide into this pleasure? If you have a regular job, the answer is pretty clear: one month a year, at best. If you own a business, again…
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The Magic of the Rotational Surface Area Formula
A formula that seems to be taught less and less in school today
The formula above gives you the surface area of the shape that is formed from rotating a curve y = f(x) around the x-axis (excluding the surfaces at each end of that shape, if any). It used to be very commonly taught in high school curricula back in the day, but I find it appearing less and less in schools nowadays. I wanted to use this article to illustrate a couple of fun and useful applications of this formula.
A sphere is formed from rotating a circle of equation x² + y² = r², where r is the radius of the circle, around the x axis. Now imagine that we wanted to find the segment of the surface area of the sphere formed from slicing it between two parallel planes.
This is equivalent to using our formula on the curve x² + y² = r², integrating between two arbitrary values a and b as per the diagram above.
Now differentiating our circle equation with respect to x, we get
which gives us
Putting this into our formula at the top of this article, we have
So this segment of the surface area is 2πr times the perpendicular distance between the planes that slice it. How nice! Of course, if we take that perpendicular distance to be the diameter of the sphere, we get the total surface area of a sphere is 4πr², which confirms the standard formula.
An ‘ice-cream cone’ can be formed by taking a circle of a certain radius, and drawing a tangent from the origin to a certain point P on the circle, and then rotating around the x-axis. This picture should help. We say that the circle is of radius a with center (c, 0) with c > a, and we draw our…
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