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