Fundamental Theorem of Calculus

Integration as Inverse of Differentation

In the previous chapter we observed that integration of a function v(t) followed by differentiation, gives back the function v(t), which we refer to as The Fundamental Theorem of Calculus:

  • \frac{d}{dt}\int_0^tv(s) ds = v(t)\quad\mbox{for }t>0.   (1)

Alternatively, The Fundamental Theorem of Calculus can be expressed as

  • \int_0^t\dot u(s) ds = u(t)\quad\mbox{for }t>0,    (2)

stating: Integration of the derivative \dot u(t) of the function u(t), gives back the function u(t). This follows from the fact that the derivative with respect to t of both sides of (2) equals \dot u(t), combined with the fact that two functions with the same derivative taking the same value for $t=0$, must coincide. Two cars traveling with the same velocitystarting at the same time from the same location will arrive at the same time to the destination. Right?

The Whole = Sum of Parts

We shall see that (2) can be viewed to express the following identity:

  • The sum  (\int_0^t \mbox{or }\sum_{m=0}^n) of the parts (du=\dot uds) =  the whole (u(t)).

Integration means summing little pieces to make up the whole. In Leibniz notation this is expressed as

  • \int_0^t \frac{du}{ds}ds=\int_0^t du =u(t)-u(0).

Elementary and profound.

Below we shall study the dependence of the integral \int_0^t v(s) ds of a given function v(s) on the time step ds, and see that it is a uniquely determined number for vanishing time step, which is approximated using a finite time step, with accuracy depending on the variation of the function f(t) with t. We will thus give a mathematical analysis of the meaning of the Fundamental Theoreom of Calculus, which we will refer to as a mathematical proof of the Fundamental Theorem.

Mechanical Analog Integrator

This experience will illustrate the role and meaning of a mathematical proof as a process of dissecting the structure and meaning of a certain mathematical statement.

Analog mechanical integrator computing the integral \int y(x)dx of a funtion y(x). Can you explain how it works?

 Read More

  •  \hyperref[chaptershortcourse]{Short Course in Calculus}
  • \hyperref[calc1d]{The Fundamental Theorem of Calculus}

To Think About

  • What could it mean to prove the Fundamental Theorem?
  • Other interpretations of: The Whole = Sum of Parts?
  • Suppose u(T)=u(0)=0. What then about \int_0^Tu^\prime (t)\, dt?

Watch

  • And as in arithmetic unpractised men must, and professors themselves may often, err, and cast up false; so also in any other subject of reasoning, the ablest, most attentive, and most practised men may deceive themselves, and infer false conclusions; not but that reason itself is always right reason, as well as arithmetic is a certain and infallible art: but no one man’s reason, nor the reason of any one number of men, makes the certainty; no more than an account is therefore well cast up because a great many men have unanimously approved it. And therefore, as when there is a controversy in an account, the parties must by their own accord set up for right reason the reason of some arbitrator, or judge, to whose sentence they will both stand, or their controversy must either come to blows, or be undecided, for want of a right reason constituted by Nature; so is it also in all debates of what kind soever: and when men that think themselves wiser than all others clamour and demand right reason for judge, yet seek no more but that things should be determined by no other men’s reason but their own, it is as intolerable in the society of men, as it is in play after trump is turned to use for trump on every occasion that suit whereof they have most in their hand. For they do nothing else, that will have every of their passions, as it comes to bear sway in them, to be taken for right reason, and that in their own controversies: bewraying their want of right reason by the claim they lay to it. (Leviathan, Thomas Hobbes)

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