Derivative volume of a sphere
WebIf you take the derivative of the volume of a sphere, $$\frac{4}{3}\pi r^3$$ you get its surface area, $$4\pi r^2$$ If you differentiate again, you get $$8 \pi r$$ Does this have any physical (or other kind of) significance, besides being $4$ times the length of a great circle on the sphere?
Derivative volume of a sphere
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WebA similar calculation is true for the derivative of the volume of a sphere. According to the definition of the derivative: Geometrically, this result is easy to understand because the … WebSep 20, 2012 · Derivative of volume 10,633 views Sep 19, 2012 13 Dislike Share Save Hunter Johnson 1.23K subscribers Subscribe Write the volume of a sphere in terms of the derivative, then find the...
WebJul 10, 2015 · When differentiated with respect to r, the derivative of πr2 is 2πr, which is the circumference of a circle. Similarly, when the formula for a sphere's volume 4 3πr3 is differentiated with respect to r, we get 4πr2. Is this just a coincidence, or is there some deep explanation for why we should expect this? calculus geometry derivatives circles WebWe can derive the familiar formula for the volume of this sphere. Finding the Volume of a Sphere Consider a cross-section of the sphere as shown. It is a circle with radius and area . Informally speaking, if we “slice” the sphere vertically into discs, each disc having infinitesimal thickness , the volume of each disc is approximately .
WebNov 4, 2012 · If you take the derivative of the area of a circle, you get the formula for circumference. When you take the derivative of the volume of the sphere, you do not get the formula for the area of a circle. Why not? d/dr (4/3pi r^3) =4pi r^2 d/dr (pi r^2)= 2pi r Answers and Replies Nov 4, 2012 #2 Homework Helper 1,388 12 WebTherefore, given the volume formula V n [R] for n-dimensional spheres, we can determine the formula V n+1 [R] for (n+1)-spheres as follows. Also, it's clear that the volume of an n-sphere must be proportional to R n, so for …
WebNote: The volume of a sphere is given by V = = Helium is pumped into a spherical balloon at a rate of 2 cubic feet per second. How fast is the radius increasing after 3 minutes? …
WebJan 30, 2024 · Since we will be taking the derivative with respect to time, we will need to treat V and r as functions of time rather than variables. In order to do this we will need to use the chain rule. So, taking the … how to renew expired driving licenceWebJan 17, 2024 · The surface area of the sphere is determined by the size of the sphere. The size is based on the radius of the sphere. The more the radius, the more will be the surface area of a sphere. The surface area of a sphere is given by \ (A = 4\pi {r^2},\) where \ (r\) is the radius of the sphere. In the formula for the surface area of a sphere, \ (4 ... nortek mighty muleWebThe volume inside a sphere is given by the formula where r is the radius of the sphere. This formula was first derived by Archimedes, who showed that the volume of a sphere is 2/3 that of a circumscribed cylinder. (This assertion follows from Cavalieri's principle .) nortek price increaseWebThe steps to calculate the volume of a sphere are: Step 1: Check the value of the radius of the sphere. Step 2: Take the cube of the radius. Step 3: Multiply r 3 by (4/3)π. Step 4: At last, add the units to the final answer. … nortek consultingWeb244K subscribers. Subscribe. 18K views 5 years ago. This video gives an informal explanation as to why the derivative of the volume of a sphere is equal to the surface area. nortek pension phone numberWebTaking the derivative of the volume does indeed give the area. Well, in the usual way. A sphere is completely characterized by its radius. The volume of a sphere is (4/3)*pi*r^3. … nortek radiant heatersWebDec 10, 2016 · That is, our volume is given by Which is the volume formula we were looking for. This same logic can be used to derive formulas for the volume of a “ball” in 4, 5, and higher dimensions as well. nortek olive branch ms