Is the surface area the derivative of the volume?
Table of Contents
- 1 Is the surface area the derivative of the volume?
- 2 What is the relationship between surface area and volume of a sphere?
- 3 What is the surface area of the sphere?
- 4 Why is the surface area of a cube not the derivative of the volume?
- 5 Why is the derivative of the area of a circle the circumference?
- 6 Why does a sphere have the least surface area?
- 7 Why is surface area of sphere?
- 8 Where does the surface area of a sphere come from?
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 relationship between surface area and volume of a sphere?
So for a sphere, the ratio of surface area to volume is given by: S/V = 3/R.
How is volume of a sphere derived?
The volume of sphere is the amount of air that a sphere can be held inside it. The formula for calculating the volume of a sphere with radius ‘r’ is given by the formula volume of sphere = (4/3)πr3.
What is the surface area of the sphere?
When we write the formula for the surface area of a sphere, we write the surface area of a sphere = 4πr2 = 4(πr2) = 4 × area of a circle.
Why is the surface area of a cube not the derivative of the volume?
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.
How the surface area of a sphere is derived?
The surface area of a sphere can be easily calculated with the help of the volume of the sphere. In this case, we should know the value of the radius of the sphere. The surface area of the sphere = 4πr2. From the formula of volume of a sphere, we can derive that, r3 = 3V/4π, or r = (3V/4π)1/3.
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.
Why does a sphere have the least surface area?
The sphere is perfectly symmetrical, and has the smallest ratio of surface area to volume of any three-dimensional shape. The internal and external forces at work within and around these structures force them to assume the shape that has the smallest possible surface area for the volume contained, which is a sphere.
Can the surface area and volume of a sphere be the same?
Area = 4 × π × r2 square units. Similarly, its volume is 4 × π/3 cubic feet = 2304 π cubic inches. So, your question really has no meaning: By choosing units appropriately, every sphere will have the same numerical value for its volume and surface area!
Why is surface area of sphere?
The Greek mathematician Archimedes discovered that the surface area of a sphere is the same as the lateral surface area of a cylinder having the same radius as the sphere and a height the length of the diameter of the sphere. The lateral surface area of the cylinder is 2πrh where h=2r .