@mr.b1130
"This is Math; we can do whatever we like." -somewhere around 5:00
@cory99998
This actually helped me wrap my head around calculus quite a bit. I understood the general concept but seeing it actually worked out this way makes way more intuitive sense than someone throwing a bunch of equations I've never heard of at me.
@yutharsansivabalan8285
Learning calculus during lockdown, Great timing
@UltimateSuperSaiyan
you can also take a triangle through a y=mx+c function and then a volume of revolution of 360° about the x axis.
@HugeRademaker
I so love your videos! Thank you. Geometric speaking: a pyramid always has a squared base. The shape with the triangular base is a tetrahedron.
@nac9880
Find the value of any shape with a rectangle: 1. Make an exact replica of the figure 2. Fulfill it with water 3. Make a rectangle replica that fits exactly all the water 4. Calculate the area of the rectangle
@ranadeep7462
IMO the expression for non inverted cone should be r/R = 1 - (h/H)
@jbw6823
When I was in high school, this was one of my fav things about calculus. I was like " why didnt you teach us this earlier!".
@guillermocasanovaaguilar8180
The exercise is very well explained. I want to make a contribution: The expression of r as a function of h comes from the fact that the triangle formed by the segments h and r is similar to the triangle formed by the segments H and R. The triangles are in the position of Thales's first theorem, so the expression comes from the similarity of triangles, and then it is automatically true that h/H = r/R. I explain this because it may be a bit difficult for someone to understand the proportional relationship between the triangles. Greetings to all.
@mendbayasgalanlkhagvadorj3097
Man keep this up, I’ve been learning a lot from this channel 👍
@musclechicken9036
you can also use H and R to form a linear function which have the x and y intercepts of the values R and H, then take the volume of revolution by integrating the linear function squared then multiplying it by PI for the radius. Though, the way you taught it is way more intuitive and beginner friendly. It helped me learn more of what an integral really is, thank you!
@snyperbro9502
At 6:38: shouldnt the formula be r/R = 1 - h/H?
@planetsharks4746
I've been watching your videos since I was 10. I've now revisited your channel again and it makes me so happy you're still making great content. You've made an impact in my life and I want to say thank you.
@ThePersonWhoKnocks
I really appreciate this video! As someone who struggled with calculus in school, this breakdown of how to find the volume of any shape using calculus is incredibly helpful. The way the author explains the concepts is so clear and easy to follow. I particularly enjoyed the explanation about how hitting infinities is when the approximation becomes exact. It's amazing how math works! I also love the comment about how the only reason to learn calculus is so you can say things like that. Overall, this video has helped me understand calculus a lot better, and I can't wait to apply these concepts in real-life situations. Thank you, Domain of Science, for another excellent video!
@kingsleychen5222
He has made a simple idea into a complicated one
@j.pesquera
This was incredibly thorough and insightful, this is basically a representation of how to actually DO calculus. Not just read a textbook, memorize formulas, and regurgitate them on an exam.
@galzajc1257
Great video and one of my favourite pieces of math. It's also very related to my last years high school research project, where i found an elementary(using just high school math) way to derive formulas for volunes of all regular polytopes and 1 formula, that directly gives volumes of all platonic and archimedean solids except the snub cube and snub dodecahedron, that were a little trickier. Moments of inertia are even more fun.
@danielrybuk1905
Instead of writing r=Rh/H i think it is simpler to write r=h×C, where C is just a constant. Integration looks the same because C is a constant, and u just plug in C=r/h in the end. I also rotated the "triangle" (the side view) 90° so it actually looks like inspecting a function over h, but that probably just personal choice. Great video, i liked it a lot, and u are right, this sort of question is very good to inspire thought and understand calc.
@herp_derpingson
I remember doing this back in high school. It was magical when I did it the first time. I did it for torus too then verified the answer with Wikipedia.
@joeymorangarza