What is the formula for calculating the surface area of a sphere using calculus?

In summary, the conversation discusses the calculation of the area for a 3D sphere using the formula for surface of revolution. The attempt at a solution involves dividing the sphere into slices and integrating to find the area, but it is determined that this method only calculates the area and not the surface area. The conversation also briefly mentions the possibility of using calculus to prove the formula for the surface area of an n-sphere.
  • #1
springo
126
0

Homework Statement


Calculate the area for 3D sphere.

Homework Equations


I know there's this formula for surface of revolution:
[tex]A=2\pi\int_{a}^{b}f(x)\sqrt{1+ f'(x)^2}\:\mathrm{d}x[/tex]

The Attempt at a Solution


I thought of dividing the the sphere into slices, each of which contains a ring.
The length of each ring is [itex]2\cdot\pi\cdot r[/itex], with [itex]r=\sqrt{R^2-x^2}[/itex].
We could then integrate:
[tex]\int_{-R}^{R}2\pi\sqrt{R^2-x^2}\:\mathrm{d}x=4\pi\int_{0}^{R}\sqrt{R^2-x^2}\:\mathrm{d}x=\pi R^2[/tex]
But this is not correct so there must be something wrong...

PS: Just out of curiosity, is there any way to prove the formula for the surface are of an n-sphere using calculus? (the one with Γ)
 
Physics news on Phys.org
  • #2
The integral which you computed is (obviously) for area based on your answer. This is because you're taking a whole bunch of rings with an infinitely small width and summing them up from 0 to R. Geometrically think of it as taking a ring and fitting successively smaller rings inside of it until the point at which all the rings together resemble a single solid. That is why you are computing the area instead of surface area.

http://en.wikipedia.org/wiki/Hypersphere see the portion on volume.
 
Last edited:

What is the formula for calculating the surface area of a sphere?

The formula for calculating the surface area of a sphere is 4πr², where r is the radius of the sphere.

Why is the surface area of a sphere important in science?

The surface area of a sphere is important in science because it allows us to calculate important physical properties such as volume, density, and pressure. It is also used in various fields such as physics, chemistry, and engineering.

How is the surface area of a sphere related to its volume?

The surface area and volume of a sphere are closely related. The volume of a sphere is equal to 4/3πr³ while the surface area is 4πr². This means that the ratio of surface area to volume is constant for all spheres, making it a useful tool in many calculations.

Can the surface area of a sphere be negative?

No, the surface area of a sphere cannot be negative. Since surface area is a measure of the exterior of an object, it is always a positive value. However, a sphere can have an imaginary or complex surface area if its radius is a complex number.

What are some real-life applications of the surface area of a sphere?

The surface area of a sphere has many practical applications, such as calculating the amount of paint needed to cover a spherical object, determining the amount of material needed to manufacture a spherical container, and estimating the amount of air resistance experienced by a spherical object in motion.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
237
  • Calculus and Beyond Homework Help
Replies
3
Views
488
  • Calculus and Beyond Homework Help
Replies
1
Views
416
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
826
Replies
14
Views
939
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
837
Back
Top