College of Engineering, Pune
Engineering Mathematics II
What Integrals are not
Normally, the study of a new concept should begin by saying what it is. In the case of integrals, it is necessary to begin by saying what they are not. Just as limits are often confused with the functional values (even though the two are conceptually quite different) and monotonicity of a function is often confused with the sign of its derivative (even though the definition of monotonicity does not even remotely involve any limiting process, much less derivatives), integrals are confused with anti-derivatives even though conceptually the two have nothing to do with each other. But then again, the reasons for these confusions are noteworthy. The reason that x→c
lim f (x) is confused with f (c) is that
the functions we encounter are generally continuous and for such functions the two are indeed equal. Similarly, even though monotonicity of a real valued function is defined independently of derivatives, the latter are often instrumental in proving monotonicity (through Lagrange’s MVT).
The reason integrals are confused with antiderivatives is similar. To define an integral b
f (x)dx you do not need the concept of derivatives. But evaluating this of the form
integral straight from its definition is possible only in a few cases. In the rest, the most standard method to evaluate this integral is to first find an antiderivative of f (x) (i.e. some function F (x) such that F (x) = f (x) for all x ∈ [a, b]). In fact, once this is done, b
all we have to do is to evaluate F at the end-points a and b. The integral
f (x)dx is
simply F (b) − F (a). This is a very basic theorem of calculus, called, quite appropriately, the Fundamental Theorem of Calculus. It needs a proof and the proof is far from trivial. But for those whose interest is only in ‘how’ rather than ‘why’, the integrals become synonymous with antiderivatives, much the same way that a bungalow which is rented and maintained by a tenant and bears his nameplate is often treated by the passers-by as his own bungalow, since they are really not interested in who the true owner is and the terms of the agreement by which the bungalow was rented. What All Integrals Are
Now that we know what the integrals are not, let us see what they are. In essence they are limits of certain sums associated with a function. So they resemble the infinite series in certain respects. But there is an important difference. We have studied only ∞
the ordered summation. Thus, in defining the sum
an , we first let Sn be the n-th
partial sum, i.e. Sn = a1 + a2 + . . . + an . Then we take the limit of Sn as n tends to infinity. Here for every n we have only one partial sum and the next partial sum, viz. Sn+1 is obtained from Sn by adding just one more term, viz. an+1 . In the unordered summation, the things are more complicated because now Sn is not just a single entity but a whole class of numbers obtained by taking the sum of n terms in all possible ways. The definition of an integral is more akin to the unordered summation. Given a function f : [a, b] −→ IR, we form certain sums called Riemann sums and the integral b
f (x)dx is the limit of these sums. In taking this limit, the number of terms has to tend to infinity. But that is not all. Individually the terms must tend to zero. Thus, figuratively, an integral is the sum of an infinite number of infinitesimally small terms. In this sense an integral is like an indeterminate form of the ∞ × 0 type. But this is only a formal resemblance and l’Hˆopital’s rule is of little help in defining an integrals. (It is sometimes useful in evaluating an integral. But it can always be byepassed.)
Practical illustrations of Riemann sums
The manner in which these Riemann sums are formed and the purpose for which they are formed can best be illustrated with a couple of examples. Suppose we want to estimate the volume, say V , of the total rainwater that falls on India. Let A be the area of India. To measure the rainfall, let us put a rain gauge at some place in India and let K be the rainfall at this selected site. Then as a very crude approximation, V ≈ KA
The trouble, of course, is that the rainfall varies a lot from place to place, especially for a large country like India. Let us say that m and M are respectively the minimum and the maximum rainfall in India. Then we have m ≤ K ≤ M and so both V and KA lie in the interval [mA, M A]. Therefore the error of the approximation in (1) is at most (M − m)A and this is all we can say about its estimation.
To improve the rainwater estimation, we divide the country into regions R1 , R2 , . . . , Rn with areas A1 , A2 , . . . , An respectively. We select some spots, one in each region and let K1 , K2 , . . . , Kn be the rainfalls at these spots. ...