Weird double integral. Please help

In summary, the conversation is about a double integral involving partial differentials and 'dT'. The person asking for help is struggling to understand how the partial differential and 'dT' disappeared in the second integral. Another person suggests using identities involving the fundamental theorem of calculus to evaluate the integral. They also provide a link for further explanation.
  • #1
racnna
40
0
Weird double integral. Please help!

its from thermodynamics...but i don't think you really need to understand thermodynamics to figure out what math trick they used to get from the first integral to the second integral
http://img833.imageshack.us/img833/833/intek.png [Broken]


i have been looking at this equation for hours and cannotfigure out how that partial differential and the 'dT' just disappeared!
 
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  • #2


Hey racnna and welcome to the forums.

Are you familiar with identities involving the fundamental theorem of calculus?
 
  • #3


hey chiro...no I am not..or maybe i have just forgotten...i just googled but can't seem to find any useful info...can you please explain this identity or link me to a place that explains it? thanks!
 
  • #4


racnna said:
hey chiro...no I am not..or maybe i have just forgotten...i just googled but can't seem to find any useful info...can you please explain this identity or link me to a place that explains it? thanks!

they evaluated the dt integral using the fact that [itex]\int_a^b \partial _x f(x,y) dx = f(b,y)-f(a,y)[/itex]

you might find this helpful;
http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign
 
  • #5


Hi there,

I understand your frustration with this double integral. It can be quite tricky to understand at first glance. However, as a scientist, I can assure you that there is a logical explanation for the disappearance of the partial differential and 'dT'.

Firstly, it's important to note that this is a common technique used in thermodynamics called "integrating by parts". Essentially, it allows us to simplify integrals by manipulating the variables involved.

In this case, the partial differential and 'dT' disappear because they have been integrated out. This means that they have been accounted for in the overall integral. This is similar to how we integrate a function with respect to a variable - the differential disappears because it has been accounted for in the integral.

I hope this helps to clarify the concept for you. If you have any further questions, please don't hesitate to ask. Keep up the good work in your studies!
 

1. What is a double integral?

A double integral is an integral that involves two variables, usually denoted as x and y. It represents the area under a surface in a three-dimensional space.

2. What makes a double integral "weird"?

A "weird" double integral is one that is difficult to solve using traditional methods, or one that involves unusual functions or limits. It can also refer to a double integral with unexpected or counterintuitive results.

3. How do you solve a weird double integral?

The best approach to solving a weird double integral is to break it down into smaller, more manageable integrals. This can involve using substitution, partial fractions, or other techniques. It may also be helpful to graph the function to gain a better understanding of the integral.

4. Why are double integrals important in science?

Double integrals are important in science because they allow us to calculate the volume, mass, and other properties of three-dimensional objects and systems. They are also used in many mathematical models and equations that describe natural phenomena.

5. Are there any real-world applications of weird double integrals?

Yes, there are many real-world applications of weird double integrals, especially in physics and engineering. For example, they are used to calculate the center of mass of an object, the moment of inertia of a rotating body, and the work done by a force on an object moving in a curved path.

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