Volume of a Frustum

Volume of the Frustum of a Cone. Here I calculate the volume of the frustum of a cone using the disk method from calculus. I also present a way of doing it without using any calculus, by using the formula for the volume of a cone. This is a very classical exercise in section 6.2 in Stewart's calculus textbook.

[drpeyam]

Insert comment about how you did it faster by doing pi h/R-r integral from r to R x^2 dx, because you can just place the cone horizontally.

[utkarshsharma]

Any binomial expansion with R and r is never gonna be the same again

[nmmm]

:)

I like the fact you began to do videos about simpler problems I can understand!

[learnology]

Thank you Dr. Peyam, it helped me at the right time.

[Julian-otcs]

I liked the cone method, but it also relies on a previously found formula for the volume of a cone, so I felt that it was less of a "complete" method in a way

[algorithminc.]

Here ... a caramel flan disappears before it can ever be measured. I think there's a physics concept built around that.
(method 1 is my vote, but it's always good to work a problem from multiple paths/methods)

[emanuelvendramini]

To get a pyramid trunk volume, just apply cavalieri's principle! (and also I prefer the integration)

[umerfarooq]

Great video, loved the bits of humour informative and fun

[mokouf]

If you forgot the formula for the volume of a cone, use calculus to find it back.

[JSSTyger]

When testing the slump of concrete, we use the frustrum of a cone and fill it in 3 layers of equal volume. R = 4in, r = 2in, and H = 12". If big R is on the bottom, what are the heights of each layer?

[alexandrezeddam]

Mmm flan au caramel. Cheers from baguetteland aka France !

[ele]

So is it a frustrum or a frustum, since I have seen both of those elsewhere too

[tmlen]

it is also possible to first calculate the area of the quadrilateral (with sides R, r, h), and then do an integral where it is extruded radially around the axis? would it make a difference that it turns faster outside than near the axis?

[aleksanderaksenov]

Sir please make extended videos on vectors and tensors

[SquidofCubes]

Have you done surface area of frustum, to get surface area element for surfaces of revolution?

[leonardmada]

Frustum in 5D:
2 balls in 4 dimensions are connected to form a frustum in 5 dimensions.
What is the volume of this frustum?

[jonathanjacobson]

So it's actually the sum of three full cones of common height h and base radii R, r & sqrt(R*r) - the geometric mean of R and r.
Perhaps someone could reveRse engineer an even more elegant solution using this fact.

[iabervon]

You could probably save a lot of writing time in the integral by doing your u-substitution for f(y) instead of expanding that in the integral.

[imnotarobot]

I kinda did it the same way (as the first method), but different way of thinking. It comes out to be exactly the same (phew..)

Triple integral (cylindrical coordinates) version:

integrate from 0 to 2pi
integrate from 0 to h
integrate from 0 to R - (R-r) * (z/h)
r * dr dz dphi

(yes, using r for the small radius and the variable for integration is incorrect, but youtube doesn't have subscripts)

This feels more elementary to me, but it requires the knowledge of the volume element in cylindrical coordinates. If I had to derive that from scratch, I'd be slower for sure :)
Volume of a rotated line segment is also kinda pretty, but again: derivation of the volume of a rotation-body takes a while.

[jamesbentonticer]

Dr Peyam now in 1080p60!
I love the integration method better :)