In which we meet another family of integrable functions, and study the connection between differentiation and integration.
- Theorem 55 Let be continuous. Then is Riemann integrable. We proved this by showing that is in the set of all such that there is a dissection of with .
- Theorem 56 (Fundamental theorem of calculus) Let be a continuous function. For , define . Then is differentiable on , and for all . We proved this by working directly with the definition of differentiability.
- Corollary 57 Let be a function with a continuous derivative. Then for all . 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 be continuous. Let be differentiable with continuous derivative, and suppose that . Take . Then . 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?
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?