Multiple Integrals 15: Volume of Sphere in Polar Coordinate

[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!

[DerUnbekannte]

dude, you're awesome, but this is not a speed competition, i think slowing down just a little would make it a lot easier to follow you.

again, i think your vids are awessome!

[DoubleDutchBust]

1:08pm Friday (EST) - Time in Haiti

[Milchi7]

Ah i see. At the beginning i didn't get the formula  r^2 + z^2 = a^2 . But it's simply that r^2 = x^2 + y^2 so that the we get the total Radius a, with every coordinate squared.

[lethe100]

I watched all of these videos..You're really good at teaching calculus!I have an exam tomorrow and you really saved my life:) Thanks a lot for sharing your knowledge and your effort..You're my teacher from now on!THANKS from Turkey..

[krisdestruction]

i dont get how you integrated the:

2r[a^2-r^2]^0.5 dr

into

-2/3[a^2-r^2]^1.5

through integration by parts (I think that's what you said)

[senox88]

thank you dony.

u helped a lot for speeding up my studies

thanks!

[andreagrimaldi]

JAJA your english is kind of funny. You've helped a lot of people very much. THANK YOU

[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.

[starrynightuk]

This is great stuff! Thank you for putting these up!

[Tsoy1984]

You are awesome!

[keledidi]

thanks a lot!

[LongShlong125]

I hate polar coordinates.