Multivariable Calculus | Triple integral with spherical coordinates: Example.

We present an example of calculating a triple integral using spherical coordinates.

[tomatrix]

Every time I see some beautiful multivariable calculus like this I am in awe. These techniques really allow us to do hard stuff so easily...

[unclesam]

I’m grateful for watching this video with such a lucid explanation. God Bless you sir

[rebucato]

Pretty sure the answer is more complicated than that...
The first integral evaluates to (4^5)/5 = 1024/5, not 2048/5
the second integral is fine
The third integral evaluates to (1-sqrt(3)/2), not 1/2, as cos(pi/6) = sqrt(3)/2, not 1/2
The answer is 512pi/5*(1-sqrt(3)/2)

Making mistakes on the blackboard is fine as long as they are corrected. However, I am shocked that there are no amendments to this video 2 years after it has been released. This shouldn't happen and I hope Penn can correct this so students would not be confused.

[betramlalusha]

I hope you are doing well and everything in your life is going great! Thank you for this! Wow!

[Martin-iwll]

I think it is necessary to check whether the top sphere actually touches the bottom cone, cause if i change the limits for y to from 0 to sqrt(1-y^2), then the answer would be very different and also 0<x<1.

[noahifiv]

Michael Penn has awesome teaching skills

[userHamza]

You explain this problem very smoothly

[seanfischler]

Thanks for the video! For some reason I had more success understanding you than my professor.

[dalibormaksimovic]

Hi. I am interested in how one can determine boundaries of integration when there is no a explicit function for z in terms of y, or y in terms of x. For instance, calculate the volume of body bounded by following surfaces: x^2+y^2 = cz, x^4+y^4=a^2(x^2+y^2) and z=0.

[maxyao]

but how can the radius be 4 if y is bounded by [0, 2]

[Carusot]

Consider the hot air balloon with equation 9x^2 +9y^2 +4z^2 = 100.
The temperature in the hot air balloon is given by the following function: f(x, y, z) = 18x^2 + 18y^2 + 8z^2 − 20

• Convert the regular area into appropriate spherical coordinates.
• Calculate the volume of the balloon.
• Calculate the average temperature in the Balloon

[walter]

Maybe somebody can help me out: I know you can see the bounds for theta on the board, but is there an algebraic way to derive them from the equations and the xyz bounds ?

[sarykhalaf]

how to determine exactly what’s the equation is????

[jaineshmachhi]

Coolest way to end the video saying" Good now it's a good place to stop"😂😂

[majidrazavi]

Excelant i realy enjoyed god bless you

[ThAlEdison]

In this problem becomes x^2+y^2=4, which means that the intersection of the cone and sphere are at the same as the limits on the x-y plane, but if the bounds on x had been 0, sqrt(1-y^2), then you would've had a spherical section on top of a cylinder on top of a cone. I'm not 100% sure how I would convert the equation if that was the case. It may require changing it to two separate integrals, one where phi goes from 0 to csc^-1(4) and rho goes from 0 to 4, and a second where phi goes from csc^-1(4) to pi/6 and rho goes from 0 to csc(phi).

[unconscious]

4^5 is not 2048 and and cos(pi/6) is not 1/2