Analysis I: Lecture 22

In which we meet another family of integrable functions, and study the connection between differentiation and integration.

  • Theorem 55 Let f : [a,b] \to \mathbb{R} be continuous.  Then f is Riemann integrable.  We proved this by showing that b is in the set A of all t \in [a,b] such that there is a dissection \mathcal{D} of [a,t] with S(f, \mathcal{D}) - s(f, \mathcal{D}) \leq \epsilon (t-a).
  • Theorem 56 (Fundamental theorem of calculus) Let f : [a,b] \to \mathbb{R} be a continuous function.  For t \in [a,b], define F(t) = \int_a^t f(x) \mathrm{d}x.  Then F is differentiable on [a,b], and F'(t) = f(t) for all t \in [a,b].  We proved this by working directly with the definition of differentiability.
  • Corollary 57 Let g : [a,b] \to \mathbb{R} be a function with a continuous derivative.  Then \int_a^t g'(x) \mathrm{d}x = g(t) - g(a) for all t \in [a,b].  We proved this using the Fundamental theorem of calculus and the constant value theorem.
  • Corollary 58 (Change of variables for integration, or integration by substitution) Let f : [a,b] \to \mathbb{R} be continuous.  Let g : [\gamma, \delta] \to \mathbb{R} be differentiable with continuous derivative, and suppose that g([\gamma, \delta]) \subseteq [a,b].  Take c, d \in [\gamma, \delta].  Then \displaystyle \int_{g(c)}^{g(d)} f(t) \mathrm{d}t = \int_c^d f(g(x))g'(x)\mathrm{d}t.  This was a corollary of the Fundamental Theorem of Calculus.

Understanding today’s lecture

You might find it helpful to consider the statements of the various results today for some explicit examples of functions.  Does this version of integration by substitution/change of variables match up with your previous understanding?

Further reading

There are many ‘fundamental theorems’ in maths.  How many do you know?  In addition to the Fundamental Theorem of Calculus, you have met the Fundamental Theorem of Arithmetic, and perhaps the Fundamental Theorem of Algebra.  Perhaps fewer of you have yet met the Fundamental Theorem of Galois Theory or some of the other ‘fundamental theorems’ that Wikipedia chooses to list.  (By the way, if you don’t know the story of Galois then it’s worth reading the biography that I’ve just linked to, or indeed any other biography of him.)

Here are links to biographies of mathematicians mentioned on the examples sheets but not in lectures (so I haven’t previously linked to their biographies): Jensen, Stirling, Wallis (a Cambridge mathematician),

Preparation for Lecture 23

Can you prove the formula for integration by parts using the Fundamental theorem of calculus?

Our next task will be to find a form of Taylor’s theorem in which the remainder term is an integral.  How might we do that?


7 Responses to “Analysis I: Lecture 22”

  1. Patrick Says:

    Unless I’ve missed something, it might be worth clarifying that in the proof of FTC, we used the fact that z was the least upper bound in saying that “by making delta smaller if necessary, we can find x in the interval [z-d/2, z+d/2] such that x is in A” – because z is the *least* upper bound, there must be a smaller member of A within distance delta/2, otherwise z-d/2 would be an upper bound. (That confused a couple of us at the time.)
    Also, we didn’t run over the case sup A = a; in this event we can’t use the interval [z-d/2, z+d/2], but I think [z, z+d/2] works fine, and the dissection D we’re working with is just {a}, with D’ = {a, a+d/2}.

  2. Joshua Says:

    Is there a difference between “Let g be differentiable with continuous derivative” and “Let g be continuously differentiable”? The latter terminology was introduced earlier in the course but you seemed to avoid using it today…

  3. theoremoftheweek Says:

    Patrick, I think that you’re referring to the proof that continuous functions are Riemann integrable rather than to the Fundamental theorem of calculus. But thanks very much for your comment: it’s always good to hear how someone has clarified something for themselves.

  4. theoremoftheweek Says:

    Joshua, I wondered whether to use the phrase ‘continuously differentiable’, but decided not to because I defined it so long ago and hadn’t used it much (if at all) since then. It would indeed have meant the same thing!

  5. Daniel Says:

    To add to what Patrick said (and similarly I apologise in advance If I have misunderstood or otherwise made a mistake):

    In my notes I copied down something like “by making delta smaller if necessary, we can assume z – δ/2 ∈ A” which is true, but since we then go on find a dissection between a and z + δ/2 if we need to reduce δ to 0 it’s not very useful.

    (I think this is in addition to the z = a case, though it happens to be true for that; I also appreciate that dealing with this just requires (I think?) declaring another variable to keep the old δ, but still).

  6. theoremoftheweek Says:

    Daniel, thanks for your comment, I did slightly gloss over this point in the lecture. A couple of people asked me about this at the end of the lecture on Friday. It might indeed be the case that we don’t want to look at z - \delta/2, but in this case z \in A so we can take a ‘good’ dissection of [a,z] and then extend to a dissection of [a,z+\delta/2] instead.

  7. Analysis I: Lecture 23 | Theorem of the week Says:

    […] Expositions of interesting mathematical results « Analysis I: Lecture 22 […]

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