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!
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
the cone phi=pi/4 the spheres rho=1 and rho=2 let's just take the top one for now. V=int[0,2pi](int[0,pi/4](int[1,2](rho^2•sin(phi))drho)dphi)dtheta V=2pi•int[0,pi/4](sin(phi))dphi•int[1,2](rho^2)drho V=2pi•(-cos(phi))|[0,pi/4]•(rho^3/3)|[1,2] V=14pi/3•(1-1/sqrt(2))
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
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!
@@saggetarious97 I want to believe, but after 2 different tries at calc 1 with 2 different teachers, and 4 more book after that, I'm a bit more than skeptical about my ability to do so.
If you rotate half a meridian that connects north to south (phi = pi) through an angle of theta = 2*pi, then you sweep the entire surface of the sphere for a radius between 0 and R
Imagine holding a plate in your hand and you spin it until it's upside down. During that half-flip, one half of the plate will trace out the top of the sphere and the other half of the plate will trace out the bottom. An example on a lower dimension: If instead of having a simple radius, you had a double-sided lightsaber radius, you'd only need 1pi rotation to form the complete sphere
@@ezxd5192You rotate the radius in 2 dimensions first to make the circle, which then covers both sides of the sphere at once so only needs to be flipped upside down like the plate in his example.
@@robertlezama1958 You can not throw equations on the board without explaining where they come from. You have to use scaffolding and spiraling when teaching.
The disk method version: ua-cam.com/video/VBR7boMYaN4/v-deo.html
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!
I love how I started my school calc with your videos and now I will get to do college with your calc 3 videos!
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
Area is crazy
@@tfg601 True i suppose. Better change it o volume
the cone phi=pi/4
the spheres rho=1 and rho=2
let's just take the top one for now.
V=int[0,2pi](int[0,pi/4](int[1,2](rho^2•sin(phi))drho)dphi)dtheta
V=2pi•int[0,pi/4](sin(phi))dphi•int[1,2](rho^2)drho
V=2pi•(-cos(phi))|[0,pi/4]•(rho^3/3)|[1,2]
V=14pi/3•(1-1/sqrt(2))
@@maxvangulik1988 Yea that's right. Want something even better?
@@allmight801 i cant even solve the previous one but sureeee
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
I am so blessed to have bprp making calc 3 videos right when I am studying to clep it in july/august
3:27 jacobian or geometry to get the internal part of the integral
I like how you say sephere! I really do.
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!
Here's a challenge/suggestion: evaluate the Jacobian of hypersphere coordinates.
Good example, problem. Krista King also has good points on this topic
I do not understand every concept on the video yet
Finally I managed to make u a video on triple integration
My goodness. I wanna know how they actually come up with the actual formulas.
I think it's because to get to the other half you can just rotate within the theta axis
Thank you !
Lezgo another upload
this is very nice
Nice
THAT WAS COOL
nice !
Seeing stuff like this makes me wish I was capable of learning calculus, it's fairly neato.
Everyone is capable of learning calculus as long as they want it enough! 😁
@@saggetarious97 I want to believe, but after 2 different tries at calc 1 with 2 different teachers, and 4 more book after that, I'm a bit more than skeptical about my ability to do so.
Why is phi not from 0 to 2pi?
If you rotate half a meridian that connects north to south (phi = pi) through an angle of theta = 2*pi, then you sweep the entire surface of the sphere for a radius between 0 and R
Imagine rotating a circle in pi radians and the visualization of the rotation, which is like rendering the sphere
Imagine holding a plate in your hand and you spin it until it's upside down. During that half-flip, one half of the plate will trace out the top of the sphere and the other half of the plate will trace out the bottom.
An example on a lower dimension: If instead of having a simple radius, you had a double-sided lightsaber radius, you'd only need 1pi rotation to form the complete sphere
@@BackflipsBenis there a reason for why it works "double sided" while theta doesn't work like that and thus we used 2pi for theta?
@@ezxd5192You rotate the radius in 2 dimensions first to make the circle, which then covers both sides of the sphere at once so only needs to be flipped upside down like the plate in his example.
BlackPenRedPenBluePen :D
i didnt understand where the p^2 * sin(phi) inside of the integral came from?
jacobian
Why is the differential ρ^2sinθ dρdθdφ? (the geometry)
ua-cam.com/video/uL_yq733CTY/v-deo.html
Cool I know two other way of proof
Now do the volume of an N-sphere
See the 1d ,2d ,3d and make the generalization to n
my dumb physicist brain cannot comprehend theta being longitude and phi being latitude
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.
What would you suggest instead?
@@robertlezama1958 You can not throw equations on the board without explaining where they come from. You have to use scaffolding and spiraling when teaching.
@@roninkegawa1804 I kinda see your point. I just like having these videos, so I want to support and encourage folks to make them.