However, often we need an antiderivative F on a closed interval [ a , b ].
The definition then specifies how to extend the above natural situation. To describe the required behaviour of F at a and b we use the second condition, that F be continuous on [ a , b ]. By the previous remark we know that due to differentiability, F would be continuous on a , b anyway, so the continuity condition actually just adds requirement of one-sided continuity at endpoints of our closed interval.
Indeed, F is continuous on the real line, so it is also continuous on 0,13]. As a matter of fact, these two functions are antiderivatives to 3 on any interval, typically we would use open or closed ones; the half-open interval in this example was used just to show how it would work. This example shows several interesting things. First of all, how did we get these antiderivatives? We guessed using our experience with derivatives.
Unlike the procedure of differentiation, where we had a reliable algorithm and could differentiate any function that comes our way, the antiderivative is a different story. We will get to it later. The second interesting thing is that we can have more antiderivatives to the same given function. The following theorem says that adding a constant to an already known antiderivative is the only way to get other antiderivatives:.
Let F be an antiderivative of f on some interval I. We can also consider this situation from a geometric standpoint. If we find such an F and then add C to it, we are simply moving the graph up or down, therefore the slopes stay the same and the shifted function also satisfies the condition. In the converse ii we are given a function f prescribing slopes of tangent lines of graphs of two functions, F and H. Imagine that we try to draw these two graphs.
We start at their left endpoints and begin drawing both of them simultaneously.
Since the two graphs have the same tangent lines at corresponding points, it means that they rise and fall in the same way; that is, we have to move the pencils in a parallel way. There is no notation to express directly the fact that some F is an antiderivative of f. However, there is a notation for the set of all antiderivatives on a given interval. We denote it by. This is called the Newton integral of f on I. Since the above theorem tells us exactly how the set of all antiderivatives looks like, we usually describe this set - the Newton integral - in the following way: If F is some antiderivative of f on an interval I , we write the set of all antiderivatives of f on I as.
Note that the notation is actually wrong. Since the antiderivatives form a set, the proper notation should be. However, this seems like too much writing, therefore people prefer the incorrect but easier notation. As long as we remember that the answer on the right hand side is a set or, to put it another way, any function of the given form, where for C we can put an arbitrary constant , we should be fine. This is a serious error, which in simple problems may look formal and like a nitpicking, but in applied problems this can be quite serious.
Unless the interval is somehow determined by the problem, we always try to put the largest interval possible. This is called the domain of the integral. We determine it by intersecting the domain of the integrated function with the domain of the antiderivative we find, perhaps removing some points where the antiderivative is defined but has some problem with its derivative.
In the above example, the domain would be the whole set of real numbers.
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Since the domain is the largest interval on which we can integrate, we can use the given answer also on "subintervals". It is clear from the definition that if F is an antiderivative of f on some interval I and J is an interval that is a subset of I , then F is an antiderivative of f on J. The fact that there are many antiderivatives also has another consequence that one has to keep in mind.
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Although we call the resulting function anti derivative, the procedure does not undo the differentiation process. So if we start with a function F , find its derivative f and then find an antiderivative of f , we need not obtain F again. If we use the Newton integral notation, this question of reversing does not even make sense. The outcome of the integral procedure is not one function but a set of functions, so it cannot be equal to the one given function that we started with and differentiated.
On the other hand, if we start with a function f , find an antiderivative F and then differentiate it, we end up with f again. This follows directly from the definition of an antiderivative. With the Newton integral notation this becomes a bit unclear: How do you differentiate a set of functions?
But if we adopt a convention for a moment that this means differentiating all functions from the given set and making it into a new set, we can write not precisely, but it captures the spirit :. Recall that when we introduced the integral notation, we started like this: "If F is some antiderivative of f on I In fact, as we will see later, some functions do not have an antiderivative, in other words, the integration procedure fails!
Introduction to Integration
But before we get to it, we look at some properties of an antiderivative:. Theorem linearity of integral. This theorem is actually very easy to prove. The second statement is equally obvious. This notation is easier to use than the language of antiderivatives when it comes to actual integration. For instance, using our experience with derivatives we guess and easily write that. In the problem we used a different variable.
Just like in the ur?
We just have to be careful not to change it by a mistake during calculations. A good habit is to check at the end that our answer has the same variable as the question. Note also that a sum of two constants whose values are arbitrary numbers is just one number, again arbitrary. Another good habit is to check that we got the right answer.
Toni Haastrup and Meryl Kenny present the different strands of institutionalism and how they were applied to EU integration so far. Then they turn to Feminist Institutionalism as a means of completing the older institutionalisms by explaining how feminist perspectives help to answer questions on asymmetric power relations. Sabine Lang and Birgit Sauer review the relatively new concept of the politics of scale as a possible theoretical approach to EU integration.
Politics of scale would help to investigate EU integration regarding its layered and intersecting arenas along the three dimensions space, access and agency, and reach. Such a perspective would not only produce a better understanding of the processes of EU integration in general, but it would also furthermore allow addressing these aspects as part of critical feminist theory-building. In the last chapter, Gabriele Wilde reviews the diverse and often gender-blind research on civil society and the public sphere.
Following up this criticism, she recommends developing a more comprehensive, feminist consideration of the political allowing understanding gender relations as constitutive for EU integration and political mobilisation. Thankfully, all book chapters follow the same structure and thereby allow for easy access. First, the particular theory is laid out covering the varieties of each, followed by detecting gaps from a gender perspective and how these could be solved, before presenting research or case studies already combining the theory and gender studies and suggesting future joint ventures.
Through this systematic structure, the editors managed to provide an excellent literature review of every theory permitting to use the volume like an encyclopaedia. Each chapter ends with a collection of discussion questions, essential readings and further references that make the book a useful resource for those teaching EU studies or courses on gender equality. The chapters as a whole are all successful in putting forward links between EU integration theories and gender studies and — in suitable cases — in carving out overlaps and similar theoretical foundations.
This specification comes in handy for imagining how to practically transfer which theory for which kind of research question keeping in mind the core aspects. An exposed point in this regard is undoubtedly the sensitivity for questions on power and power relations: the power to shape, the power to participate, the power to decide. Unanimously, the edited volume was written from the perspectives of gender studies and — without question — all authors are experts in the theory presented.
Introduction to Integration
The conclusion honours this perspective; however, more of such chapters or as alternative joint sections would have satisfied the wish for more dialogue. The edited volume, at least, brilliantly illuminates how such a dialogue deepens and widens perspectives for both sides. Laurel Weldon eds. Oxford: OUP. Wiener, Antje and Diez, Thomas eds. She obtained her Dr. Her research focuses on European integration, gender equality policies, public policy and transnational social movements.
She is co-editor of the German feminist journal Femina Politica. Before you download your free e-book, please consider donating to support open access publishing. E-IR is an independent non-profit publisher run by an all volunteer team. Your donations allow us to invest in new open access titles and pay our bandwidth bills to ensure we keep our existing titles free to view.
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