Calculus can strike fear in the hearts of mortal men. Really, it should not be that way. It’s not as bad as all that. But I don’t care to learn it you might say. That’s your prerogative, of course. Yet, I must say you may be the poorer for it. If that is your stand, however, all is not lost. You can memorize certain equations and still get a lot from it.
Consider the example of deriving the surface area of a sphere by differentiating (a big but harmless word) its volume.
If the volume is V, based on radius R, and the surface area is S, this is written,
dV/dR = S
The equation is read, “The derivative of the volume (with respect to its radius) is its surface area.”
Interestingly, the above equation, while it explains what you are going to do, is not what you need to memorize, so if you choose to, you can forget what we’ve written above. To move on, if we consider a function (an equation) of one variable, such as the radius, then
F = X n, and
The Derivative is n X (n-1).
That’s the formula to memorize! D = n X (n-1). As an example, for a sphere, the volume V is,
V(sphere) = 4/3 πr 3
Applying the format of our memorization equation, we derive
V(sphere) = 3 (4/3) πr (3-1) = 4 πr 2, or
S(sphere) = 4 πr 2.
Let’s try another geometrical figure to see if our formula works for that, also. We’ll try
The Formula for the Volume of a Cylinder
The formula for the volume of a cylinder is,
V(cylinder) = πr 2 h
Then we calculate,
S(cylinder) = 2 πr (2-1) h = 2 πrh.
But no! Here is the catch. This is not the total surface area! What? Our formula is correct as far as it goes. This is the surface area of the vertical surface of our cylinder. It does not include the surface area of the top or the bottom. But doesn’t this really make sense? Imagine two cylinders with the very same radius, standing side-by-side. If one is twice as tall as the other, its volume should be twice as much, no? But is its surface area twice as much? No. Only on its vertical surface area is twice as much. There is no “h” in the formula for the area of the ends. The ends of both cylinders have identical surface area, namely two times πr 2 for each end, giving us a total of 2 πr 2. So the total surface area of a cylinder is,
S = 2 πrh + 2 πr 2
This can also be written,
S = 2 πr (h + r)
So keeping a sharp mind to recognize the adjustments that might need to be made, such as its ends in the case of the cylinder (or one end in the case of a cone), the single most useful formula for the common man to remember, derived from the calculus, is,
Derivative = n X (n-1)
The World of Math Online – “Surface Area Formulas”