Friday, July 4, 2014

Area: Interesting Results

Area 1 (see figure 1 above)

This is a simple rectangle.

A = r * h


Easy, right? Suppose that two of the sides are not straight lines but sinusoidal curves, like in Figure 2 below.

yTOP = sin(2 * π * x/r) + h
yBTM = sin(2 * π * x/r)

Left boundary: x = 0
Right boundary: x = r

Area:

r
∫ yTOP - yBTM dx
0

r
∫ sin(2 * π * x/r) + h - sin(2 * π * x/r) dx
0

r
∫ h dx
0

r * h


Hmmm. Let's try something with curves shaped parabolically, like in Figure 3.

yTOP = -x^2 + r * x + h
yBTM = -x^2 + r * x

Left boundary: x = 0
Right boundary: x = r

Area:

r
∫ yTOP - yBTM dx
0

r
∫ -x^2 + r * x + h - (-x^2 + r * h) dx
0

r
∫ h dx
0

r * h

Interesting that the same result is obtained in all three cases, A = r * h.

Keep in mind that these are specific shapes. Who said math isn't fun? :)

Eddie


This blog is property of Edward Shore. 2014


2 comments:

  1. Great post. Lots of material in one tidy package. Much nourishment:)

    ReplyDelete
  2. I think this generalizes. If Ytop = f(x)+h and Ybottom=f(x) then the area is still r*h. This should work for any f(x) that's defined over the interval, even one that has no integral itself... I think....

    ReplyDelete

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