Surface Area of a Sphere without double integral

In summary, the conversation discusses different methods for calculating the surface area of a sphere without using a double integral. These methods include using shell integration, the First Theorem of Pappus, and finding the derivative of the volume function.
  • #1
cwbullivant
60
0
Is it possible to come up with a derivation of the surface area of a sphere without using a double integral? Most of the ones I've found seem to involve double integrals;

For example, this was given as the "simplest" explanation in a thread from 2005:

[tex]S=\iint dS=R^{2}\int_{0}^{2\pi}d\varphi \int_{0}^{\pi} d\vartheta \ \sin\vartheta[/tex]

I was thinking about using shell integration for it, but as I recall, shell integration and solids of revolution deal only in volumes, not surface areas (This was by far my weakest area of Calc II, FWIW).

I'm going to be doing double integrals fairly soon, but I wanted to know if there was a more simplistic method so I wouldn't have to wait until then.
 
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  • #2
You can do surface areas in single variable calculus, but they are just as ugly as they are in multivariable. The reason is that instead of using dx you have to use ds where ds = arc length = ##\sqrt{1+ (\frac{dy}{dx})^2} dx##

The general formula for surface area is ##SA = \int 2 \pi y ds##. So a formula for surface area of a unit sphere would be ##\int_{-1}^{1}{2 \pi y \sqrt{1+ (\frac{dy}{dx})^2}} dx## where y is the half circle of radius 1.
 
  • #3
You might try the First Theorem of Pappus applied to a semi-circular arc.
 
  • #4
SteamKing said:
You might try the First Theorem of Pappus applied to a semi-circular arc.

Wow, that's a super cool theorem. Never knew about that.
 
  • #5
Pappus' theorem is a cool way.

Another way goes like this. For a sphere, let V(r) be the volume, and S(r) be surface area (as a function of radius).

Then V(r) = V(1)r^3 and V'(r) = S(r). Therefore, S(r) = 3V(1)r^2.

So if you can calculate the volume of a sphere (via the method of disks for example), then you can get the surface area this way.
 

1. What is the formula for finding the surface area of a sphere without using a double integral?

The formula for finding the surface area of a sphere without using a double integral is 4πr^2, where r is the radius of the sphere.

2. How does this formula differ from the one used for finding the surface area of a sphere with a double integral?

The formula for finding the surface area of a sphere with a double integral is 2π∫rds, where r is the radius of the sphere and ds is an infinitesimal element of the sphere's surface. This formula is more complex and requires knowledge of calculus.

3. Why would someone want to find the surface area of a sphere without using a double integral?

Using the formula 4πr^2 to find the surface area of a sphere without a double integral is a simpler and more straightforward method. It is also helpful for those who do not have a strong background in calculus.

4. Can this formula be used to find the surface area of any size sphere?

Yes, the formula 4πr^2 can be used to find the surface area of any size sphere as long as the radius of the sphere is known.

5. Are there any limitations when using this formula to find the surface area of a sphere?

The only limitation when using this formula is that it only applies to perfectly spherical objects. If the object is not a perfect sphere, then this formula will not give an accurate result.

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