Estimating area under a curve in log scale

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Discussion Overview

The discussion revolves around estimating the area under curves presented in a log-log scale without having the equations for the curves. Participants explore various methods for estimation, including graphical techniques and numerical approaches, while addressing the challenges posed by the logarithmic scales.

Discussion Character

  • Homework-related
  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in estimating the area under the curves using rectangles and trapezoids, noting inconsistencies in their results.
  • Another participant suggests defining the area more clearly, proposing to find the area bounded by the curves and the x-axis within specific limits.
  • A different approach is proposed involving physical methods, such as cutting out the area under the curves from printed images and weighing them to estimate the area.
  • One participant recommends dividing the area into simple shapes to simplify the estimation process, emphasizing that more effort leads to better estimates.
  • Another participant humorously claims that the area is infinite due to the nature of the log scale, suggesting it may be a trick question.
  • A later reply advises ignoring parts of the graph below a certain threshold, suggesting that these contribute negligibly to the overall area and emphasizing the importance of accurately estimating the peak values.
  • One participant suggests redrawing the graph in linear scales to facilitate area calculations using rectangles.

Areas of Agreement / Disagreement

Participants present multiple competing views on how to approach the estimation problem, with no consensus on a single method or solution. Some methods are more physical or graphical, while others are analytical or numerical.

Contextual Notes

Participants note the challenges of working with log scales, including the arbitrary nature of the x-axis and the implications for area calculations. There is also mention of potential errors in estimation due to the logarithmic representation.

matan100
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Homework Statement


Hi guys, I'm stuck on this question in exam revision:
I need to estimate the area under the following curves (I don't have the equations of the curves)
http://img819.imageshack.us/img819/3249/question.jpg

The curves are magnitude diagrams in the log-log scale.


Homework Equations





The Attempt at a Solution


I've tried a general estimation (using rectangles and trapezoids) treating it as a linear graph but I come up with inconsistent answers. I've searched everywhere looking for a method to estimate area in the log-log space.

I'm completely stumped. Thanks for the help
Cheers
 
Last edited by a moderator:
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Do you mean the area under the curves as shown in the graph? You need a proper description of the region whose area you want. For example, it might be "For each function, find the area of the region bounded by the graph of the function and the x-axis, between x = 10-3 and x = 102.

I would estimate each area using vertical rectangular strips. You could use trapezoids, but it would be a bit more work. For each rectangular strip you need the rectangle's height and width. You can read the height from the function's graph, and the width of each strip varies as x increases from left to right. For example, between x = 10-3 and x = 10-2, each tickmark is 1/1000 or .001. Between x = 10-2 and x = 10-1, each tickmark is 1/100 or .01.
 
Print the image on two pieces of paper, cut out the paper under each curve down to the X-axis and weigh each piece.

Cut a third piece of paper along the X-axis, along the right and left edges of the graphed area, and along the 10^0 horizontal line to give you a reference weight. You'll have to consider how to account for the log scales.

If you don't have a sufficiently sensitive scale, paste the images on cardboard before cutting.
 
A really quick estimate would be given if divide each area into simple shapesy sum up the individual areas of them. I did a quick outline (really quick) of the solid curve. The red lines are the estimates. The more time you put into it the better the estimate.

Screenshot2010-06-11at123841PM.png
 
The x-axis is arbitrary on a log scale. The area is infinite. It's a trick question :-p
 
You can also ignore anything below 10^-2 because this is less than 1 % of the peak value.
Given the other errors, 1% is an unlikely precision.

Especially concentrate on that peak at the top. Try to get that right because the values in there are much higher than the rest of the graph and so the errors would be greatest there also.

Redraw the graph converting both scales to linear values, otherwise you can't take rectangles and calculate areas.
 

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