MA1102 Calculus Tutorial 8

Topics Covered:

Anti-derivative. Change of variable formula (substitution). Riemann integrals. Limits as integrals via Riemann sums. Use of Substitution. Fundamental Theorem of Calculus. Evaluation of area under a curve.

Textbook: Chapter 8, Chapter 9

Anti-derivatives are just the opposite counterparts of derivatives. The elementary technique of finding anti-derivatives is to tabulate known derivatives of polynomial functions, trigonometric functions, exponential functions, logarithmic functions and use these information as we use a look-up table. For the more difficult functions we use the substitution or change of variable formula to extend our 'look-up table' and to provide a more efficient means to get the answer. The 'look-up table' is in the form of integral formulae for polynomial functions and trigonometric functions for the moment. More formula are to be added later as more functions are defined later on. It will be a little harder and may be impossible to obtain a closed integral formula for piecewisely defined function. Examine the concept of Riemann sum and remember the Riemann integral in this case is a complicated kind of limit of expressions depending on the partition and the choice of values in each subinterval of the partition. We give a heuristic definition of the Riemnann integral. But in practice, the proofs of the results involving Riemann integrals would require a more practical characterisation of integrability. This is where we ask for your patience. Deeper results involving uniform continuity and the notion of supremum and infimum are the terms that may be used in some of the proofs. For the serious minded go through the proofs given in the text book. Later when you have learnt the concepts of infimum, supremum and uniform continuity, return to fill up where you have left off.

The change of variable formula or substitution is just an application of the chain rule. To use this tool the integrand must be put in the form of the derivative of the desired substitution times another function, where substitution can be performed. To use the Riemann sum to calculate the limit of an infinite series will involve rewriting the partial sum of the series as a Riemann sum of a function. Of course you would have to work out the function, the partition, the lower and upper limits. How does this work? We are using only the special Riemann sum, either a sum of the form . Assuming that f is continuous and therefore Riemann integrable on [a, b] and so the limit of the special Riemann sum with respect to regular partitions will converge to the Riemann integral of f from a to b. (Try proving this fact.)

The Fundamental Theorem of Calculus provides a recipe to produce an anti-derivative when we needed one, in terms of Riemann integrals. As much as this gives the answer to finding anti-derivative, that is, every continuous function is the derivative of some function, it opens up another question: if f is differentiable, what can we expect f ' to be like? Can we recover f at least up to a constant from f ' ? The answer is no, surprisingly. So continuity, differentiability and integrability are rather different concepts. Differentiable functions are not necessarily better behaved than continuous ones. The following is a deep result: If f is differentiable, then f ' must be continuous at some point. The fundamental Theorem of calculus bridges Riemann integrals with differentiation or more precisely, with anti-derivatives. This provides the bulk of application in everyday science or engineering. It is indeed an extremely useful and powerful tool. Remember the condition for the Fundamental Theorem of Calculus calls for a continuous function f . Then where F is any function such that F'(x) = f (x) for all x in [a, b]. Here is a less well known result. If we assume that F'(x) is Riemann integrable (and F' (x) need not be continuous) on [a, b], then we have . This is Darboux theorem. The proofs in Chapter 9 are well worth studying, especially that of the Fundamental Theorem of Calculus. Some of the proofs are very difficult. You will observe how ideas, properties and the nature of the mathematical objects are used.