Lecture 27 Definite integrals

Lecture 27: Definite integrals 2013 Only by admitting an infinitesimal unit for observation – a differential of history, that is, the uniform strivings of people – and attaining to the art of integrating them (taking the sums of those infinitesimal quantities) can we hope to comprehend the laws of history. Leo Tolstoy, War and Peace III.3.1 1 Introduction Today we define the definite integral, which is nothing but a precise notation for the exact area under a curve. I will discuss two ways to conceive of what this means: one classical (from the first 200 or so years of the existence of calculus) and one modern (from the last 150 years or so). We will compute a few definite integrals in cases where nothing is required but elementary geometry. We will then discuss the notion of signed area, which is a mild generalization of “area under a curve” that is more useful in most applications, and state some formal properties. The next lecture concerns the fundamental theorem that is used to actually compute definite integrals by hand. The reference for today is Stewart §5.2. 2 The definitie integral: notation The following is the usual notation for what is called a “definite integral.” b Z f (x)dx a It means, roughly speaking: “the area under the curve y = f (x) between x = a and x = b.” Caveat: this definition is a bit misleading when f (x) has negative values. I’ll discuss this later, in the section “signed area.” For the time begin, I’ll just consider positive functions so that this basic intuitive definition works. For example, consider this expression. Z 3 xdx 0 The graph y = x is a straight line of slope 1. So this notation refers to the area shown. 1 3 This is just a triangle with base and height 3; its area is 21 3 · 3 = 9 2 = 4.5. Therefore 3 Z xdx = 9/2 0 3 The classical viewpoint: sums of infinitesimal quantities Rb The classical viewpoint on the notation a f (x)dx explains why the notation is written the way it is. Remember that in the previous lecture, we saw that you could approximate the area under f (x) from a to b by summing the ares of n rectangles, each of width ∆x and height xk (for some value xk in the interval). f (xk ) ∆x 3 As the number of subdivisions becomes smaller and small, this approximation will become better and better. f (xk ) ∆x 3 2 f (xk ) ∆x 3 The classical viewpoint is: every single one of these approximations is incorrect, because they all choose a unit (the width) totally arbitrarily. So instead, we should take an infinitely small (infinitesimal) width, written dx, and sum up infinitely many rectangles that are all infinitesimally thin. f (xk ) dx 3 Now, just as the notation for a sum of finitely many rectangles of some finite width is written this way, n X f (xk )∆x k=1 the classical viewpoint thinks of the following notation as meaning the sum of infinitely many rectangles, each of the infinitesimal width dx, for values of x from a to b. Z b f (x)dx a Note the similarity between these two notations: both have a symbol means to means sum (the greek letter S, Σ in the first case, and a stylized roman S in the second case). According to the classical view, summing the infinitesimal rectangles will produce no error at all, because they eliminate the arbitrary choice of unit that would plague usual rectangular approximations. The Tolstoy quote that began this lecture (which Tolstoy says in a passage where he speaking about historical analysis, but uses math as an example) speaks to this older viewpoint on integrals: they were formerly regarded as a sum of infinitely many quantities (differentials) that are all infinitely small. This is ∆y dy , meant as an infinitesimal version of ∆x , is meant to evoke arithmetic totally analogous to how the notion dx with infinitesimal quantities. For this reason, the field we now call calculus was originally called “infinitesimal calculus,” roughly meaning “the practice of calculating with infinitesimal numbers.” The first textbook on calculus (written by L’Hôpital) was, in fact, titled “Analysis of the infinitely small.” 3 This classical perspective is still widely used colloquially, especially in physics. I think it is the best intuitive way to think about integrals. Nonetheless, it does invite various philosophical objections about whether it could possibly be sensical; it was mainly to answer these objections that the modern perspective (described below) emerged. 4 The modern view: limits dy to be something Recall that when we studied derivatives, we wanted to define the slope of the tangent line dx like the slope of a secant line to an infinitely near point. To make this precise, however, we used the notion of limits: we said that for any finite distance ∆x, we could calculus the slope of the secant line to appoint ∆y . These numbers actually settle down and approach a limit as ∆x shrinks to 0. So ∆x units away as ∆x rather than speaking of “infinitesimal differences,” we just spoke about limits of secant lines, and defined dy ∆y = lim . dx ∆x→0 ∆x The same perspective leads to the modern way of defining integrals. Instead of speaking directly about “a sum of infinitely many things,” we just speak of the sum of n things, and then take a limit as n grows to infinity. Thus, following Bernhard Riemann, we can make the following modern definition. Z b f (x)dx = lim a n→∞ n X f (xk )∆x k=0 (where, for a chosen value of n, ∆x stands for (b − a)/n and xk stands for a + k∆x). The modern view has two primary merits. Rb • Philosophical: this provides a clear, unambiguous definition of the notation a f (x)dx that avoids the ontological vagueness involved in “summing infinitesimal quantities.” • Computation: Riemann sums can be computed explicitly with computers. Therefore we can use the modern view to approximate integrals that would be impossible evaluate explicitly (all that remains is to answer the question: how large should n be for a good approximation? This sort of problem is the subject of a huge part of applied mathematics). 5 Some examples However you think of the definite integral (as a sum of infinitesimal pieces, or as a limit of sums), you can always just think of it as a way to measure area. In some simple cases, integrals can be evaluated just be measuring area using basic geometry. Here are three examples. R5 Example 5.1. Consider the integral 0 f (x)dx, where f (x) is the piecewise linear function shown below. The area under this curve consists of two parts: the area under the setment from (0, 5) to (2, 0), and the area under the segment from (2, 0) to (5, 3). The first is a triangle of area 21 5 · 2 = 5, while the second is a triangle of area 21 3 · 3 = 9/2. So the total area under the curve is 5 + 9/2 = 19/2 (or 9.5). 4 Z 5 f (x)dx = 19/2 0 Z 1p Example 5.2. Consider 1 − x2 dx. 0 √ The graph of y = 1 − x2 is a semicircle of radius 1. To see this, notice that it just consists of all points on the curve x2 + y 2 = 1 where y ≥ 0. The area described by this integral will just be the area under this semicircle from x = 0 to x = 1, which is a quarter-circle, as shown. The whole circle has area π. Therefore Z 1 p π 1 − x2 dx = . 4 0 Rx Example 5.3. Let x be a variable. Consider the integral 0 tdt. If we graph the function f (t) = t, we will obtain a straight line of slope 1. So the area under this curve from 0 to x (where x > 0 for now) is the area of a right triangle with base x and height x. Therefore Z x 1 tdt = x2 2 0 Notice that 21 x2 is actually an antiderivative of f (x) = x. This is not a coincidence, and this is a topic will be at the center of the next lecture. 6 Signed area In this section, we extend the meaning of the notation negative values, and when a > b. 6.1 Rb a f (x)dx to two new cases: when f (x) could have Negative values of f (x) First, consider the case where f (x) is sometimes negative. For example, consider the graph of sin x from x = −π to x = π. In this case, part of the graph is below the axis (vertical shading), and part of it is above the axis (horizontal shading). In cases like this, we use the following convention. 5 Area below the x axis counts negatively towards the value of the definite integral. Another way to put this is: Z b f (x)dx = (area above) − (area below) a In fact, this is completely in line with what will happen if we compute using Riemann sums: when f (x) is negative, we’ll be adding terms f (xk )∆x to the sum, which will be negative. In case of sin x, we can see that the part below is exactly the same shape as the part above, so without knowing anything else about this integral, we can conduce that Z π sin xdx = 0 −π This convention (area below the axis is subtracted) is also the best to use in most applications. For example: Example 6.1. Suppose that p(t) measures the profit per day of a company at a time t. If p(t) > 0, this means the company is making profit; if p(t) < 0 this means that it is losing money. Then 10 Z p(t)dt 0 perfectly captures the total profit of the company from time t = 0 to time t = 10: the positive parts add up the profit (on days when the company made money), and the negative parts add up the total loss (on days when the company lost money). For these reasons, we sometimes say that the definite integral computes signed area: it counts some area as positive, some area as negative, and adds it all together. Z 2 Example 6.2. Compute xdx. −1 Solution. The graph of y = x is shown, with the area between the graph and the curve shaded (vertical shading will be negative area, horizontal shading will be positive area). The positive part is a triangle with area Z 2 1 1 1 1 · 1 = . Therefore xdx = 2 − = 3/2. 2 2 2 −1 6.2 1 22 · 2 = 2, while the negative part is a triangle with area “Backwards” intervals Rb Next consider the situation where we write a f (x)dx, but a > b. For example, what should we mean by R0 xdx? The answer is given by the following convention. 2 Z b Z f (x)dx = − a f (x)dx b 6 a Therefore, re-express an integral R 0 we canR always R 2 as one whose endpoints a and b satisfy a < b. For 2 example: 2 xdx = − 0 xdx, which is −2 because 0 xdx is the area of a triangle with base 2 and height 2. What is the reason for this convention? There are two good reasons for it. Rb • Theoretical. You can think of a f (x)dx as begin a sum of infinitesimal quantities f (x)dx, where x is moving from a to b, and dx measures its change. Now, if a is actually greater than b, this means that x is moving backwards, so dx will be negative. So the resulting sum is the same as if x moved forward from b to a, except that everything is made negative. Alternatively, if you use the definition of Riemann sums, then if b < a then ∆x = b−a n will be negative, and so the Riemann sum will end up counting everything with a minus sign. • Practical. This convention keeps many identities and computational techniques about integrals neat and tidy. To explain what I mean by neat and tidy, consider for example the following identity. Z b Z Z c f (x)dx f (x)dx = f (x)dx + a c a b This identity is visually obvious if a < b < c: we’re computing one area by breaking it into two pieces (the area above [a, b] and the area above [b, c]). By taking the sign convention above, this identity is universally true. For example, if a < c < b, then identity above says that to compute the area above [a, c], we can compute the area above [a, b] and then subtract the area above [c, b]. Identities like this are very important in computations, so it is very useful to make sure that they are universally true, so that you don’t have to worry about strange edge cases when applying them. 7