Geometric understanding of integration / surface area of sphere

Click For Summary
SUMMARY

The discussion centers on the geometric understanding of the surface area of a sphere, specifically addressing the confusion surrounding the integral 2∫0R 2πr dr and its relation to the formula for the surface area of a sphere, 4πR2. The integral calculates the area of a disc, not the surface area of a sphere, as it does not account for the varying angles (θ) necessary for spherical geometry. The correct approach involves considering the sphere's slices, which require integration over the angle θ rather than just the radius r.

PREREQUISITES
  • Understanding of basic calculus concepts, including integration and differentiation.
  • Familiarity with geometric shapes, particularly spheres and circles.
  • Knowledge of the formulas for the volume and surface area of a sphere.
  • Ability to interpret and manipulate integrals in a geometric context.
NEXT STEPS
  • Study the derivation of the surface area of a sphere using spherical coordinates.
  • Learn about the concept of differential elements in integration, particularly in relation to geometric shapes.
  • Explore the relationship between volume and surface area through calculus, focusing on the use of derivatives and integrals.
  • Investigate the application of integrals in calculating areas of more complex shapes beyond simple discs and spheres.
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus and geometry, as well as anyone seeking to deepen their understanding of the relationship between integration and geometric properties of shapes.

Saraphim
Messages
46
Reaction score
0
Hi everyone,

I've browsed around the forum a bit and found that others have had the same problem as me, however, none of the answers help me a lot, so I thought to post a more specific question, I hope you don't mind.

I'm having a problem with the surface area of a sphere, probably because I'm extending "backwards", ie. if the derivative of sphere volume is sphere surface, why is the integral over the circumference of a circle not the surface area of a sphere? More formally, why is this:

2\int_0^R 2\pi r dr \neq 4\pi R^2

I know plenty of other ways of obtaining the surface area of a sphere, also through integration, but I simply lack the geometric understanding to show me why this particular integral does NOT describe the surface area of a (hemi)sphere.

Can anyone enlighten me? Bear in mind, I am by no means very good at math. I know a bunch of tools, and how to use them, but my understanding of them is vague at best, and that is what I am trying to rectify.

Thank you in advance. :-)
 
Physics news on Phys.org
Hi Saraphim! :smile:

2πrdr is the area of a ring of inner radius r and outer radius r + dr.

So ∫0R 2πrdr is the area of a whole disc of radius R.

To find the surface area of a sphere, you would divide the sphere into tiny slices, but they would all have the same r, wouldn't they? :wink:

(the expression inside the ∫ would be the area of a typical tiny slice)

They would have different θ instead, and you would integrate over θ.
 
It didn't even occur to me to stop and consider the meaning of the differential. :blushing:

Thanks a lot. :)
 

Similar threads

Undergrad Calculating the surface area of a sphere using dA
  • · Replies 33 ·
2
Replies
33
Views
7K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
3K
High School Question about volume of a sphere
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Double integrals - do areas cancel?
  • · Replies 24 ·
Replies
24
Views
4K
Undergrad Result of this integral in large Lambda limit
  • · Replies 12 ·
Replies
12
Views
3K
MHB How to Find the Surface Area of a Pringle Using Double Integration?
  • · Replies 1 ·
Replies
1
Views
3K
High School A question about one of Archimedes' works
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Why is the limit of θ from 0 to π in the formula for surface area of a sphere?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Gabriel's Horn, Inside vs. Outside Surface area
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Re-derive the surface area of a sphere
  • · Replies 3 ·
Replies
3
Views
2K