Volume of a sphere with a triple integral

We will use a triple integral with spherical coordinates to show that the volume of a sphere with radius a is 4*pi*r^3. This is a very cool problem that you need to see for your Calculus 3 class.

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[cdkw]

I love how I started my school calc with your videos and now I will get to do college with your calc 3 videos!

[allmight]

You know what is even more interesting? It's finding the volume of a figure that is bounded by z=sqrt(x^2+y^2), x^2+y^2+z^2=1 and x^2+y^2+z^2=4 with a triple integral

[primaryendo]

10 years ago when i was in electrical engineering a teacher gave this problem on a test and i got it right. I've forgotten how to do calculus now, but i still find it so cool!

[cringy-year-old]

I am so blessed to have bprp making calc 3 videos right when I am studying to clep it in july/august

[alexdefoc]

When i was in 10th grade i started to find you and many other math channels. Since then i became a math addict and been going to events and it was fun. Math is an enjoyment now thank you so much!. Making it always interesting, these small things, we encounter randomly when we learn something hard, then suddenly the connection form and we can learn faster and easier and have more fun

[terawattyear]

Yes neat! Way cool. I ended my studies of Calculus with dropping out of my Calculus 2 class - 45 years ago or so. However, I pretty much can follow what you are showing here, and it is indeed elegant. Reminds me of the rotation of equations to yield solids which I really loved. Thanks!

[yash]

3:27 jacobian or geometry to get the internal part of the integral

[Siccmann]

Here's a challenge/suggestion: evaluate the Jacobian of hypersphere coordinates.

[Iomhar]

I like how you say sephere! I really do.

[danmike]

Good example, problem. Krista King also has good points on this topic

[Luigi-hnhu]

I do not understand every concept on the video yet

[Sgth]

My goodness. I wanna know how they actually come up with the actual formulas.

[Alphamatics]

Finally I managed to make u a video on triple integration

[lauraprates]

I think it's because to get to the other half you can just rotate within the theta axis

[TiagoSilva-ycbe]

i didnt understand where the p^2 * sin(phi) inside of the integral came from?

[NeedsMorePants]

Seeing stuff like this makes me wish I was capable of learning calculus, it's fairly neato.

[unflexian]

my dumb physicist brain cannot comprehend theta being longitude and phi being latitude

[roninkegawa]

If this is how you teach math, you must have confused students who all fail your class. This was a perfect example of how not to teach math.