MT101-03, Area Notes. We briefly mentioned the following triumvirate of integration that we ll be studying this semester:

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1 We briefly mentioned the following triumvirate of integration that we ll be studying this semester: AD AUG MT101-03, Area Notes NC where AUG stands for Area Under the Graph of a positive elementary function, AD stands for AntiDifferentiation, and NC stands for Net Change. We also briefly discussed why the AUG problem is perhaps the most motivated one for us at this point, if only from an agricultural or geodetical point of view. So we ll study this problem first. Precisely, the problem is to find the area under the graph of any positive elementary function over any given interval where the function is defined. 1 The general consensus nowadays is that a mathematical definition doesn t say what something is, but instead defines how it behaves. So here is how mathematicians have come to see area and shape behave. A shape is a subset S of the coordinate plane R 2 with a welldefined area, which we write as A(S), and shapes and area interact as follows: (1) Area is never negative. (2) Every box is a shape, and the area of a box is its length times its width. (3) Every subset of a shape with zero area is another shape with zero area. (4) A translation, rotation, or reflection of a shape is another shape with the same area. (5) Subtracting one shape from another shape yields another shape: e.g., B A FIND ME But what do we mean by area? A - B A B is a shape in the above figure, since it s the difference of two boxes, which are shapes.

2 2 (6) The union of a sequence of disjoint shapes is another shape, whose area is the sum of the areas of its disjoint parts. That is, if S 1,S 2,... is a sequence of shapes such that S i S j = φ for distinct i,j, then U = S 1 S 2, their union, is itself a shape, and A(U) = A(S 1 )+ A(S 2 )+. THAT IS ALL WE NEED, with regards to assumptions about area and shape. Here are some examples: A point is a box whose length and width are 0. So it is a shape (with zero area). - A line segment is a box whose width is 0. So it is a shape (with zero area). A box is a shape by definition. A box with a missing side is a box minus a line segment. So it is a shape. [Questions] Is a circular disc a shape? What about a parabolic region? As it turns out, we can reduce these questions to our initial objective. For a circular disc is the union of two half-discs, and a half-disc is the region under the graph of y = r 2 x 2. Similarly, it is not too hard to see that a parabolic region is the difference of a trapezoid and the region under the graph of a quadratic function. We will call the Region Under the Graph of a positive function f over an interval [a,b] the rug of f over [a,b]. The area of the rug is what we re looking for; it s the area under the graph of the function over [a,b]. This is important enough to merit a formal definition and special notation. Suppose R is the rug of a positive function f over an interval [a,b], and suppose A(R) is well-defined (i.e. suppose R is a shape). Then we say the following things: f is integrable over [a,b] (or from a to b). the (definite) integral of f over [a,b] (or from a to b) is A(R). b f(x)dx = A(R). a So we may focus on the following two objectives: [AUG 0] Determine when a rug is a shape, and [AUG 1] Find a way to calculate exact expressions for rug areas/definite integrals. Our approach to both is similar to that for derivatives: approximate the area with a sequence of ever better approximations, and take the limit of the sequence of areas.

3 3 Now, we want a general method following the above approach that will apply to every elementary function. We need to approximate the areas with shapes as general as possible. But the shapes can t be too complicated, or they ll be too difficult to work with! We have to balance simplicity and generality. This balance is achieved by flat skylines, which we ll just call skylines. (Our text uses finite skylines, whose areas are called Riemann sums.) [Example] [Example] [Example] A flat skyline, or just skyline if flatness is understood, is the union of a sequence of mutually friendly boxes whose lowest sides all lie along the same line, usually the X-axis; and where two boxes are friendly if their intersection has zero area. (We call such boxes friendly since if two boxes of land are not friendly, the owners may dispute over who owns the intersections.) That is to say, a skyline is just a bunch of rectangles glued together on one side of the X-axis. [Proposition] Skylines are shapes. [Sketch Proof] All skylines are disjoint unions of sequences of boxes with pieces of sides missing. And boxes with pieces of sides missing are shapes. Observe the examples, and meditate. Q.E.D. [Homework] Is a right triangle a flat skyline? Justify your answer.

4 4 Intuitively, skylines can imitate rugs very well. For instance, here is a sequence of skylines approximating a half-disc ever better: And so on. Here is another such sequence: And so on. Why do we intuitively feel like these are good approximations? Their boundaries get ever closer...? Their difference 1 gets ever smaller in area...? Now, the first bullet is complicated to define properly, and at any rate is not what we re interested in. We are interested in area. So the second bullet, perhaps, is what we should use. But if we use only the second bullet as a definition of good approximation, we will encounter pathological sequences of skylines (which I do not care to show you here; cf. appendix). 1 The (symmetric) difference of two sets A and B is (A B) (B A).

5 5 Instead, we refine the second bullet into two similar notions: approximation from inside and approximation from outside, which the example sequences already illustrate for you. The reason these notions work is the following intuitive theorem: [Theorem] If one shape S lies entirely within another shape R, then A(S) A(R). [Proof] R S is a shape, and A(R S) 0, since area is never negative. Also R = (R S) S and (R S) and S are disjoint. Therefore A(R) = A(R S)+ A(S) Q.E.D. A(S). [Homework] Show that if A(R S) = 0, then A(R) = A(S). Therefore, if we assume that R is a rug with a well-defined area that is to say, if we assume that the rug R is a shape then A(S) A(R) for any skyline S lying strictly within R. Furthermore, we expect to be able to approximate A(R) arbitrarily well with the area A(S) of a skyline S lying entirely within R, as in the first sequence. Geometrically, this means that if we plot the areas of all the skylines lying within R, there will be plots arbitrarily close to A(R) to the left of it, and none to the right of it: A(R) This is achieved precisely when A(R) is the smallest number bigger than or equal to all the areas A(S) of all the skylines S lying strictly within R. That is, A(R) is the supremum of all those areas. However, we don t yet know for sure if R is a shape or not. Therefore, we make the following definition: The inner area inn(r) of a rug R is the supremum of all the areas of skylines within R. Similarly, again assuming that a rug R is a shape, any skyline S containing R has A(S) A(R). And, again, similarly we expect that the area A(R) is the biggest number smaller than all the areas of skylines containing R, as in the second sequence. A(R) That is, A(R) is the infimum of all those areas. But again, we don t know yet if any given rug R is a shape or not. So we make the following definition 2 : The outer area out(r) of a rug R is the infimum of all the areas of skylines containing R. 2 One may define a more general notion of inner and outer area for arbitrary subsets of the plane, using general rectangular blobs. But we need not do that for this course.

6 6 Of course, if a rug is a shape, we expect that its inner and outer areas will be the same. As a matter of fact, it turns out that this is precisely the condition guaranteeing that a rug is a shape! [Theorem] A rug R is a shape precisely when its inner and outer areas are equal to the same number, say a. In that case, A(R) =a. [Proof] Not at all difficult, but its length would distract us, I m afraid, from our main narrative. Also, it seems pretty obvious/believable anyway. So I ve written a proof of it for my sanity s sake, but I ve relegated it to an appendix for your sanity s sake. Q.E.D. This is great! We now have a way to determine whether a rug is a shape or not, which completes objective [AUG 0] right? Well, we still don t have a way to tell from a formula for a function whether its rug is a shape or not i.e., whether f is integrable or not. So it s probably not fair to say we ve completed objective [AUG 0] yet. Also, our putative technique for calculating definite integrals involves taking limits of sequences of areas of skyline approximations. If we wanted to stop proving theorems right now and calculate some definite integrals with these raw definitions, we would have to [A] find a sequence of skylines inside our rug whose areas approach a value L in from below, [B] find a sequence of skylines outside our rug whose areas approach a value L out from above, [C] develop notation to express the areas, [D] make sure our sequences have nice, palatable expressions in our notation by choosing the sequences nicely, and, finally, [E] show that L in = L out. As you might imagine, this is [1] more trouble than it s worth, [2] primarily of historical interest, and [3] not particularly motivating except as a thought-experiment. So instead we ll career onwards towards our goal: the Fundamental Theorems of Calculus.