(March 15, 2016 at 3:51 am)robvalue Wrote: Yes, they are. Such equations represent a relationship between them.
This is why simple addition will never work. To add together infinitesimals, you have to use integration rather than summation. It's a method of continuous adding.
It does seem extremely bizarre to begin with!
When you sum up infinitesimals, you'll always be summing and infinite amount of them, throughout any particular range. For example, between x=0 and X=1, I will cover every value between and there will be a dx for each one.
The easiest way of visualising integration is to look at a graph, such as y = x^2. You use integration to find the area under the graph, between any given point. What you are actually doing is splitting the area up into tiny trapeziums. Think of a very thin one, with the X coordinates almost identical, going up to meet the graph. If you split up the interval into a thousand sections, you'll have a thousand trapeziums, giving a pretty close approximation of the total area. The flat tops of the trapeziums just won't be exactly the same as the curve of the graph.
As you allow the number of sections to approach infinity, the accuracy improves to be exact. You're summing an infinite amount, each with width dx, and the relationship between dx and the values of dA (the change in area dx causes) represent the trapezium approaching zero width.
In general dA = y dx since y is the height of the trapezium at any given point (instead of meeting the graph twice, it only meets it once at an exact point, so there is just one height measurement now)
In this case the relationship is dA = x^2 dx
So A = X^3/3 to be evaluated between any two values of X.
Thanks; good explanation.
