Multiple Integrals 15: Volume of Sphere in Polar Coordinate

We visit an age-old problem of finding the volume of a sphere but this time curiously using polar double integrals.

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Tags:

• calculus
• multivariable
• sphere
• integral
• double
• polar
• math

[bizarreworld123]

Will it be the same procedure if more than half of the sphere is above the xy-plane? E.g. if the center of the sphere is say (0,0,2) where the volume above the xy-plane is not equal to the volume below.

[donylee]

Yes, in a way. In fact, the scenario you described requires careful thinking. Intuitively, we can identify the plane where the sphere is symmetrical, namely one perpendicular to z-axis at point (0,0,2). Imposed region R on this plane to the xy-plane and calculate.

BUT (and this a big BUT!), how are you going to describe the sphere in this case above the xy-plane by z=f(x,y)? Notice that it is NOT a single function in terms of x and y. Can't be as each (x,y) gives 2 values of z. Think about it.

[Nymphetamine2791]

Could you write a triple integral with ranges: outside integral still: 0 to 2pi, middle integral still: 0 to a, and inside integral: -sqrt((a^2)-(r^2)) to +sqrt((a^2)-(r^2)) to express volume of a sphere? If you can't, how can you while using triple integrals?

[Fabri92Rssista]

"just imagine that the 2 is dead" LMAO!!

you're awesome, keep up the good work!

[hniang2]

you are very funny, looks like you are giving a speech:D great job mister:D

[RockieBurnz]

Since when Harold started teaching calculus?

[iceeqn]

great video. very helpful. :)

[siroichix]

woww thankss... you're speaking style is kinda cool... i like it..

[boredpeople100]

THANK YOU!

I was about to hammer my head to the desk trying to figure out why there was a two =)

[bunberrier]

Thanks donylee!