# Integration without an expression

• johnintheuk
In summary, the conversation revolved around finding the quantity of energy transferred between two systems using absorption and drive curves. It was suggested to integrate the curves, but since they do not follow a mathematical expression, alternative methods were discussed such as drawing and measuring the area, using a planimeter or online software, or counting pixels. It was also suggested to break the curve into smaller sections and find matching functions for each section to calculate the total integral.
johnintheuk
I'm trying to find the quantity of energy transferred between two systems. I have an absorption curve and a drive curve, but neither of these follow a mathematical expression, they're random squiggely lines.

I want to use the absorption curves to find out how much of the drive is being transferred. I know one way to do this would be to integrate the curves, but they can't really be approximated by an expression.

How else might I go about it?

I have actually thought about drawing them, then doing the old cutting out and measuring the area trick for a rough estimation. Surely there's something easier and more accurate than that though.

Maybe scan them and use something on the computer to find the enclosed area?

But there's a lot of sharp deviation on the curves, so I'd really need to cut the x-axis up into a lot of sections to get any kind of accuracy whatsoever.

Is there any online software that I can draw the curves in (dragging points), or something like that?

Depending on how it crosses the horizontal axis, you may be able to use a planimeter. Otherwise, scan or trace it into a computer and using http://livedocs.adobe.com/en_US/Photoshop/10.0/help.html?content=WS3D3EF585-502B-49d2-85FF-537E9DC25C21.html or http://www.ma.iup.edu/projects/CalcDEMma/Green/Green.html , you can calculate the area contained above the axis/below the axis and take the difference for the proper integral.
Otherwise, do it the old fashioned way: use graph paper to trace and approximate coordinates and use a numerical integration algorithm to calculate the area.

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Thanks slider, although the problem is more complex than just area. I need to reference this curve to another, and then use the second to find the percentage of the first that's being absorbed, and I need to do this along the entire length of the curves, for at least tens of points, preferably more.

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it might be easier to scan it, and then instead of integrating, just count the amount of non-white pixels over the x-axis. then to find the length of the curve, again, just take the number of columns of pixels.
percentages, can just be calculated.

Well, the whole of the line may not have a similar looking function,

But! If you break the line into tiny sub-sections and find the functions

that may match each individual little line, you can simply add all the

integrals of the tiny lines together.

Maybe this will help :)

## 1. What is integration without an expression?

Integration without an expression refers to the process of finding the antiderivative of a function without being given an explicit expression for the function. This usually involves using techniques such as substitution or integration by parts.

## 2. Why would I need to use integration without an expression?

There are many situations in math and science where we encounter functions that do not have a simple expression, but we still need to find the area under the curve or the volume of a solid. Integration without an expression allows us to solve these problems without needing to know the exact form of the function.

## 3. How is integration without an expression different from regular integration?

In regular integration, we are given a specific function and we use techniques to find its antiderivative. With integration without an expression, we do not have a specific function, but rather a general form or a set of conditions that the function must satisfy. This can make the integration process more challenging and require more advanced techniques.

## 4. What are some common techniques used for integration without an expression?

Some common techniques used for integration without an expression include substitution, integration by parts, partial fractions, and trigonometric identities. These techniques allow us to manipulate the function to a form that is easier to integrate and find the antiderivative.

## 5. Can integration without an expression be used in real-world applications?

Yes, integration without an expression is commonly used in real-world applications in fields such as physics, engineering, and economics. For example, when calculating the center of mass of a complex object or the work done by a variable force, integration without an expression is often necessary to find the solution.

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