- #1
DoubleMike
- 39
- 0
So this question has been bothering me for a very long time... but only recently have I mustered up courage to register to ask it.
Before though, let me draw a parallel analogy, so you can see where I'm coming from. When you take the volume of a region enclosed by one or more functions revolved about a line, either the method of diks or shells is used.
Let us examine the volume of a sphere. The simplest way to calculate the volume of a sphere of radius R is to use the disk method with limits of integration -R to R, in respect to either variable (due to the symmetry).
So in essence the Riemann sum is the addition of the volume of infinitely many cylinders. These have parallel sides to the axis of the sphere/integration. Though strictly speaking we are adding very small volumes, I'd rather think of it as adding an infinite number of areas. This goes well with my understanding of infinities (as I've entreated in the General Math Forum). If this is flawed however, please feel free to elucidate.
Anyway... When asked to take the surface area of a sphere, my book goes about it a very strange method indeed. It breaks the sphere up into an infinite number of partial cones, and uses the formula of their lateral surface area. I understand why this works... it makes sense. But why won't this work:
Breaking the sphere up into an infinite number of cylinders, and using the surface area formula for those (translational surface of a circle).
Here I have supplied a picture of what I mean:
http://www.geocities.com/boeclan/sphere.jpg
*(if the link doesn't work, copy and paste into address bar)
That picture goes for both the surface area and volume integrals. In the case of volume, that is the correct cross-section, whereas for surface area it is not.
Why is it that the function's slope is ignored when calculating volume, but for surface area, one must use the circuitous method described by my book?
*edit = typo
Before though, let me draw a parallel analogy, so you can see where I'm coming from. When you take the volume of a region enclosed by one or more functions revolved about a line, either the method of diks or shells is used.
Let us examine the volume of a sphere. The simplest way to calculate the volume of a sphere of radius R is to use the disk method with limits of integration -R to R, in respect to either variable (due to the symmetry).
So in essence the Riemann sum is the addition of the volume of infinitely many cylinders. These have parallel sides to the axis of the sphere/integration. Though strictly speaking we are adding very small volumes, I'd rather think of it as adding an infinite number of areas. This goes well with my understanding of infinities (as I've entreated in the General Math Forum). If this is flawed however, please feel free to elucidate.
Anyway... When asked to take the surface area of a sphere, my book goes about it a very strange method indeed. It breaks the sphere up into an infinite number of partial cones, and uses the formula of their lateral surface area. I understand why this works... it makes sense. But why won't this work:
Breaking the sphere up into an infinite number of cylinders, and using the surface area formula for those (translational surface of a circle).
Here I have supplied a picture of what I mean:
http://www.geocities.com/boeclan/sphere.jpg
*(if the link doesn't work, copy and paste into address bar)
That picture goes for both the surface area and volume integrals. In the case of volume, that is the correct cross-section, whereas for surface area it is not.
Why is it that the function's slope is ignored when calculating volume, but for surface area, one must use the circuitous method described by my book?
*edit = typo