Why is the derivative of the volume of a sphere equal to the surface area?

The rate of change of the volume of the sphere is equal to the surface area of the sphere. The outside of the paint is the new boundary of the sphere, and the inside of the paint is added to the volume. This explains why the derivative (rate of change) of the volume is the surface area (SA).

Why is the derivative of the volume of a cube not the surface area?

How about a cube with volume V=x^3? In this case the derivative is 3x^2 so it is not the surface area. Why is this? It is due to symmetry i.e. it depends on whether the volume increases symmetrically when you increase your variable such as length of side, radius, or height, etc.

Is the surface area the derivative of the volume?

The volume of a solid can be thought of as the an infinite sum of the areas of similar “shells” arranged around a single point. This means that the Volume is the Integral of the Surface Area with respect to the radius. By the FTOC, the Surface Area is the derivative of the Volume.

What is the differentiation of volume of sphere?

The volume of a sphere is (4/3)*pi*r^3. If we differentiate that with respect to r, we get 4*pi*r^2, and lo and behold – that’s the formula for the surface area of the sphere. This makes sense, because the way the volume increases is for that surface to sweep through an additional bit of radius.

What is the derivative of surface area of cube?

S(x)=6×2 and V(x)=x3 respectively, with S being the surface, V the volume and x the edge of the cube. By deriving those, we get the functions: S′(x)=12x and V′(x)=3×2 which represent the rate of change (since that’s what a derivative essentially is) of the surface and cube.

Why is the derivative of the area of a circle the circumference?

If you increase the radius of a circle by a tiny amount, dR, then the area increases by (2πR)(dR). . That is, the derivative of the area is just the circumference. This makes the “differential nature” of the circumference a little more obvious.

Is the surface area of a sphere greater than the volume?

The increase in volume is always greater than the increase in surface area. This is true for cubes, spheres, or any other object whose size is increased without changing its shape.

Why is surface area to volume ratio important in biology?

Smaller single-celled organisms have a high surface area to volume ratio, which allows them to rely on oxygen and material diffusing into the cell (and wastes diffusing out) in order to survive. The higher the surface area to volume ratio they have, the more effective this process can be.

How does the volume of a sphere work?

The volume V of a sphere is four-thirds times pi times the radius cubed. The volume of a hemisphere is one-half the volume of the related sphere. Note : The volume of a sphere is 2/3 of the volume of a cylinder with same radius, and height equal to the diameter.