Analysis I: Lecture 19

In which we define integration.

  • We focus on a bounded function f on a closed interval [a,b],  Definition of a dissection \mathcal{D} of the interval [a,b].  Definition of the upper sum S(f,\mathcal{D}) and lower sum s(f,\mathcal{D}) of f with respect to \mathcal{D}.
  • Lemma 45 If \mathcal{D} and \mathcal{D}' are dissections with \mathcal{D}' \supseteq \mathcal{D}, then S(f,\mathcal{D}) \geq S(f,\mathcal{D}') \geq s(f,\mathcal{D}') \geq s(f,\mathcal{D}).  We proved this by adding one point at a time to \mathcal{D}.
  • Lemma 46 Let \mathcal{D}_1 and \mathcal{D}_2 be any two dissections.  Then S(f,\mathcal{D}_1) \geq S(f,\mathcal{D}_1 \cup \mathcal{D}_2) \geq s(f,\mathcal{D}_1 \cup \mathcal{D}_2) \geq s(f,\mathcal{D}_2).  This was easy with the help of Lemma 45, but is crucial to making our definition of integration work.
  • Definition of the upper integral I^{\ast}(f) and the lower integral I_{\ast}(f).
  • Lemma 47 We have I^{\ast}(f) \geq I_{\ast}(f).  This was immediate from Lemma 46.
  • Definition of what it means for f to be Riemann integrable.

Understanding today’s lecture

It would be a good idea to work through the various definitions and results that we had today for some examples, to get a feel for what they mean.  Pick a function and a dissection.  (I suggest a friendly function like a constant function or f(x) = x to get started!)  What are the upper and lower sums?  Can you find the upper and lower integrals?

It would be an excellent idea to pick some examples of functions and see whether they are Riemann integrable, working directly from this definition.  Next time, we’ll encounter a handy criterion that can make it easier to determine whether a function is Riemann integrable, but this doesn’t give the same insight into how the definition works.  Before you start thinking about difficult functions like sine, I suggest that you start with simpler functions like constant functions, or step functions, or polynomials of small degree.

Riemann integrability is another good property that you could usefully add to your functions grid (even if you’re not yet in a position to decide whether many functions actually are Riemann integrable).

Further reading

This is a good topic to look up in an introductory analysis or calculus book.

Wikipedia has some exciting animations illustrating different dissections: one and another.

As I mentioned in the lecture, there are other forms of integration, such as Lebesgue integration.  This is discussed in the Part II course on Probability and Measure.

There’s a bit about the history of calculus on MacTutor.

Preparation for Lecture 20

What properties do we hope that the integral will have?  Can you prove them from the definition?

Can you think of any classes of functions that might be Riemann integrable?  Can you prove that they really are Riemann integrable?

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One Response to “Analysis I: Lecture 19”

  1. cameron Says:

    Hi Vicky,
    You’re not the only one who takes correct pronunciation very seriously!:

    Cameron

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