It makes sense now! da/dr, the ratio between a tiny change in r and the increase in area it resulted in, is simply the circumference as dr gets infinitely thin.
Thank-you. Much more elegant than integration by parts and trig substitutions to solve for a sum rectangles in just a quarter of the area. The hard way is a good exercise but the elegant way is more honest. The area is the sum of all of the circumferences.
I had a question ,is dy/dx is a ratio ?like fractions I Does dx has value? What Dx mean ? Limit delta x tend to 0 means dx is 0? It is simple question but i m freaking out for the answer can u reply immediately
Yes you can, but you have to be careful when setting your coordinate and the limits of your integration. One of them would be r1 and one would be r2. You may want to look at his video titled "Volume of a Sphere: Three Different Derivations " as he explains the idea over there abit clearer.
One thing is bothering me regarding the last part where you explained where the formula for the sphere volume comes from: 1) We chop up the sphere into very small circles just like you would chop an onion 2) Integrate circumferences of all these circles from 0 to r The thing is that when we do the chopping, we get 2 circles of the same radius for 0 to r: as we chop from 0 to r and (at this point the half onion is left) as we chop from r to 0. I thought this would be involved in your calculations. Sorry for being so tidious about my explanation. Thank you for these intuition-cultivating videos!
well we get the number pi by asking how many diameters can we fit in the circumference of a circle and that number is approximately 3.1416, and 2pir is equivalent to piD(2 * r = D) where D is the diameter. So if C / D = pi multiply both sides by D, we get piD = C which is again equivalent to 2pir.
@@maxsmith5504 consider a straight line with length D. Now imagine you rotate the line 180° about its midpoint. You'll see that the endpoints of the line trace a semicircle each, thus creating a circle. Naturally, if you increase the length of this line, you can imagine that the circle formed, will be proportionately larger, cos your angle of rotation remains constant at 180°. You also will be knowing that 180° is π radians. That in itself is the proportionality constant. The value of π can be determined using the definition of 1 radian and measuring that angle in degrees (which is something around 57.71°, maybe the decimals might be different) and comparing it to 180°. That's how you get the 3.141529....
