Why are they called differentiate & integrate ?

[physixer]

Why are they called "differentiate" & "integrate"?

I was wondering why. When we differentiate, don't we result in an equation that describes the slopes of many other equations ( + C ) ? So, we'd be "integrating" many equations, different from each other by C, into a single description.
Inversely, when we integrate something, we get many solutions ( + C ). Aren't we producing many from one? In other words, "differentiating"?

Just curious. I'm too lazy to google the original reasonings for these terminologies :P

edit: oops, this should be in General or Calculus forums. Sorry for that.

 

[tiny-tim]

hi physixer!

i think the noun "differential" originates from "difference" methods …

eg 1 4 9 16 25 …

the differences are 3 5 7 9 …

(and the second differences are 2 2 2 …)

so when we differentiate, we start with f(x+h) - f(x), which is an infinitesimal difference, which eventually was called a differential​

and the verb "integrate" comes from the (ancient greek) idea that an area or volume can be approximated by adding up all the slices, ie combining all the parts into the whole (and "integral" means "whole")

 

[arildno]

First off:

Producing the (+C)-solutions are generelly called finding the anti-derivative of a function.

The fundamental theorem of calculus says that we may accurately perform the integration process by finding an anti-derivative, and calculating the difference between the anti-derivative's function values.

The integration process on its own is to add together an infinite number of pieces, i.e, making something "whole"/integrated as tiny-tim said.

 

[physixer]

Okay, thanks!

 

[blue_raver22]

The terms "differentiate" and "integrate" were coined by German mathematician Gottfried Leibniz in the 17th century. He used the terms "differenzieren" and "integrieren" in his writings, which were later translated to English as "differentiate" and "integrate".

The term "differentiate" comes from the Latin word "differentia", which means difference or distinction. When we differentiate a function, we are essentially finding the difference between two values of the function as the input (x) changes. This gives us the slope of the function at a particular point, which is why we often refer to it as finding the "rate of change" of the function.

On the other hand, the term "integrate" comes from the Latin word "integrare", which means to make whole or complete. When we integrate a function, we are essentially finding the sum of all the infinitesimal changes in the function over a given interval. This allows us to find the total area under the curve of the function, which is why we often refer to it as finding the "accumulation" or "total amount" of the function.

So, in summary, the terms "differentiate" and "integrate" were chosen to reflect the fundamental operations of finding the slope and area of a function, respectively. They have become standard terminology in mathematics and are used to describe these operations across various fields of study, including science and engineering.