Surface area of a sphere using integrals

In summary, the sphere's volume is calculated using slices of it with infinitesimal height, but the area is calculated using the difference between the sphere's surface area and the distance from the center of the sphere to a given point.
  • #1
spec00
3
0
Hello dear colleagues!

Yesterday i was trying to proof the surface area of a sphere formula, then i got some problems. I know that something is seriously wrong in this concept, but i can't tell what exactly is wrong. Could you guys help me please?

I just thougt about a hollow sphere, then we can slice it up to little cylinders with infinitesimal height (like slicing some onion rings). If we add up these teeny weeny little parts, i thought that we could obtain the area of the sphere.

[tex]x^{2}+y^{2}=r^{2}[/tex]
[tex]\int(2*\pi*x)*dy[/tex]

[tex]2*pi*\int\sqrt{y^{2}-r^{2}}*dy[/tex]

But when i integrate over 0 to R and multiply all by 2, the result is not correct. What did i do wrong?

Thanks!
 
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  • #2
welcome to pf!

hello spec00! welcome to pf! :smile:
spec00 said:
I just thougt about a hollow sphere, then we can slice it up to little cylinders …

no, they're not cylinders (with vertical sides), they're little slices of a cone (with sloping sides), which can have a much larger area :wink:
 
  • #3
Thanks for the fast reply, tiny-tim!

Well, that's true, but I've seen a proof for the volume of a sphere using slices of it with infinitesimal height. Since the slices are not cylinders, but little cones, that deduction wouldn't get us in the wrong place too?

http://en.wikipedia.org/wiki/Sphere (Proof of the sphere's volume formula)

Thanks again! And sorry about my english, I'm a bit rusty!
 
  • #4
hi spec00! :smile:
spec00 said:
… I've seen a proof for the volume of a sphere using slices of it with infinitesimal height. Since the slices are not cylinders, but little cones, that deduction wouldn't get us in the wrong place too?

we get this question quite often! …

the error for the volume is the difference between 2πr dh and (roughly) 2π(r + dr/2) dh …

a second-order error of π drdh​

but the error for the area is the difference between 2π dh and 2π dh/cosθ …

a first-order error of 2π(1 - secθ) dh :wink:

(in layman's terms, most of the volume is in the middle, and the error is only an edge-effect, but for the area, it's all edge! :biggrin:)
 
  • #5
Wow! Now, that's absolutely clear for me.

Thanks for the huge enlightment..!
 

1. How is the surface area of a sphere calculated using integrals?

The surface area of a sphere is calculated using the formula S = ∫∫sinθ dθ dφ, where θ ranges from 0 to π and φ ranges from 0 to 2π. This double integral represents the surface area of each infinitesimal patch on the sphere, which is then added up to find the total surface area.

2. What is the significance of using integrals to find the surface area of a sphere?

Integrals allow us to calculate the surface area of a sphere with greater precision, as it takes into account the curvature of the sphere. This is especially important when dealing with non-perfect spheres or spheres with irregular surfaces.

3. Can you explain the steps involved in using integrals to find the surface area of a sphere?

The first step is to set up the double integral using the formula S = ∫∫sinθ dθ dφ. Then, we evaluate the inner integral with respect to θ, using the limits of 0 to π. Next, we evaluate the outer integral with respect to φ, using the limits of 0 to 2π. Finally, we combine the results to find the total surface area of the sphere.

4. How does the radius of the sphere affect the surface area calculated using integrals?

The radius of the sphere directly affects the surface area calculated using integrals. As the radius increases, the surface area also increases, and vice versa. This is because a larger radius means a larger circumference, which leads to a larger surface area.

5. Are there any real-life applications of using integrals to find the surface area of a sphere?

Yes, there are many real-life applications of using integrals to find the surface area of a sphere. For example, it is used in engineering and architecture to calculate the surface area of curved structures such as domes and arches. It is also used in physics and astronomy to calculate the surface area of planets and other celestial bodies.

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