The Fundamental Theorem of Calculus Part 1
If f(t) is integrable over the interval [a,x], in which [a,x] is a finite interval, then a new function F(x) can be defined as:
For instance, if f(t) is a positive function and x is greater than a, F(x) would be the area under the graph of f(t) from a to x, as shown in the figure below:
Therefore, for every value of x you put into the function, you get a definite integral of f from a to x.
The equation above gives us new insight on the relationship between differentiation and integration. If f is a continuous function, then the equation above tells us that F(x) is a differentiable function whose derivative is f. This can be represented as follows:
In order to understand how this is true, we must examine the way it works. If f ≥ 0 on the interval [a,b], then according to the definition of derivation through difference quotients, F'(x) can be evaluated by taking the limit as h→0 of the difference quotient:
When h>0, the numerator is approximately equal to the difference between the two areas, which is the area under the graph of f from x to x + h. That is:
If we divide both sides of the above approximation by h and allow h→0, then:
This is always true regardless of whether the f is positive or negative. Therefore, it embodies Part I of the Fundamental Theorem of Calculus.