PID controllers, for one. http://en.wikipedia.org/wiki/PID_controller materials with 'memory' is another: the only referece I could quickly find has to do with mechanical properties, but any consititutive relation will do. http://www.stanford.edu/~richlin1/sma/sma.html
What exactly does it mean to be a "real life application for engineering"? There are more applications for integration than can possibly be listed here. How specifically does something have to involve integration to suit your purposes? For example, Maxwell's equations can be written in integral form. Numerical solutions of Maxwell's equations can be directly used for a huge number of engineering applications. Integration is involved in practically every physical theory in some way. Name some type of engineering task you have in mind, and I bet someone can tell you how integration is essential, at least indirectly.
Good points. I'm not speaking on behalf of the OP, but I interpreted the question to mean "what are some physical phenomena that can be modeled using integration?" or "what are some engineering applications of integration?"
Are you joking? There's like 10 billion. Anytime you have something changing, you can use both derivatives and integration. Currently, Im using a ton of integration to model system's change in mass, energy and momentum over time or other conditions.
I am really confused by this thread. Think of all the problems you have solved using integration and think of what physical significance these problems had.
All these clever replies and not one specific example of how, during the course of say, a mechanical engineers day, what problems he would be trying to solve using a method of integration...? I'm currently learning all about these methods, and knowing to what practical purpose they are applied will help me greatly in understanding the values I arrive at. For example, when you find the area under a curve, what does it represent? There seems to be come clever people on here, now be smart.
What people are asking is for the poster of the thread to be smarter. He/she could find many examples in textbooks specific to the field of interest -- or even state the specific area of interest in the thread... note the TYPE of engineering was not even posted in the original thread, let alone a specific topic within the field -- that's why I was leaving this thread alone (perhaps hoping the original poster came back with more specification). Granted, maybe prior responders could be more polite (and some have been) but now you've butted in (withyour first post to a "real" part of PF I might add) criticizing the whole... and your post isn't really even smart. You ask about a mechanical engineer, but what is he/she in the process of designing... is the main interest of his design -- vibration, distortion under weight, or one of many types of "fluid flow" -- be it heat flow, air flow (over a wing), or water flow (over a ship's hull, through a pipe, or perhaps even groundwater flow regarding a contaminant) etc.? All these things can be either directly solved by integration (for simple systems), or some type of numerical integration (for complex systems). Note: I'm not even a mechanical engineer, but these are the things you'll often see in an introductory "boundary values" text. Then you ask about the area under a curve... The area under a curve (or for that matter its slope) can represent many things, or nothing of any real physical significance or interest at all, depending on what is plotted in the curve. To be polite, however, to the original poster and others of interest in this topic: Some areas in the past where I've done numerical integration: The first time I did numerical integration was when I was a student in a chemistry class and needed to do numerical integration to analyze some data regarding pH in a titration (forgive me that this was 15 yrs ago -- I don't have the book anymore and don't recall the details because I don't work in chemistry or chemical engineering). More recently: when I worked in the field of electro-optical engineering and wanted to design optical waveguides for certain systems, and I did some numerical analysis to estimate the power (intensity) profile of the exiting beam (this depends on Maxwell's equations, boundary conditions required within these equations, and the specific boundary profile in space created by the waveguide and the material it was made of -- I mention this because some other respondents mentioned Maxwell's equations). There was also some analysis of how the time profile of a "square pulse" would be changed as it propagated through the structure (to see if it spread too much to be above a certain signal to noise requirement -- with given types of noise being added). As an aside, numerical integration is also important in other fields (like biology -- say for population analysis or kinematic studies of cell processes, etc.) or economics, but again, those are not my particular fields.
Despite the hypocritical comment in your second paragraph, the rest of what you go on to say is exactly what I'm imagining the original poster of this thread was wanting... I only tried to assist by suggesting a mechanical engineer; however my knowledge is limited, which is why I hadn't included a specific area of interest, leaving that to the smart(er) experienced engineers on here. The examples you give show you have much experience in this area, and have applied uses of integration to many different problems, and I agree that the particular topic is important. From what I have learnt so far, it appears that you find either very general examples of how to integrate, such as in a first year calculus paper, or very specific examples of its application, such as the PID algorithm, the link to which was posted previously... there is no middle ground. I'm sure during the course of my university career I, and others here in the same boat, will learn more about integration and its applications, what the figures represent physically, and how we can use it as a tool to solve many problems in our day once we are qualified. I expect that this forum will remain a supplementary tool to assist our learning.