Surface Area and Volume of a Sphere

  • #1
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Main Question or Discussion Point

I need to know the Surface Area and Volume of a spherical ball at the origin radius a.
What I want is to evaluate the integrals at each integral.

##\oint_S dS =\int\int d? d? = 4 *\pi*r^2##

##\oint_V dV = \int_0^{\pi}\int_0^{2\pi}\int_0^a dr d\theta d\phi## = ##\frac{4}{3}*\pi*a^2##
 
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Answers and Replies

  • #2
33,722
5,419
I need to know the Surface Area and Volume of a spherical ball at the origin radius a.
What I want is to evaluate the integrals at each integral.

##\oint_S dS =\int\int d? d? = 4 *\pi*r^2##

##\oint_V dV = \int_0^{\pi}\int_0^{2\pi}\int_0^a dr d\theta d\phi## = ##\frac{4}{3}*\pi*a^2##.
Your last formula is incorrect: the volume of a sphere is ##(4/3)\pi r^3##.

Also, you don't want closed path integrals - ordinary integrals will do just fine. For the surface area, you can do this with a single integral that represents the surface area of a surface of revolution, and for the volume, you can do this by calculating the volume of a solid of revolution
 
  • #3
mathman
Science Advisor
7,839
439
I need to know the Surface Area and Volume of a spherical ball at the origin radius a.
What I want is to evaluate the integrals at each integral.

##\oint_S dS =\int\int d? d? = 4 *\pi*r^2##

##\oint_V dV = \int_0^{\pi}\int_0^{2\pi}\int_0^a dr d\theta d\phi## = ##\frac{4}{3}*\pi*a^2##
The differential for surface area is dθsinφdφ with coefficient r2, for volume is r2drdθsinφdφ.
 

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