area of specific shape based on ellipse

niksirat said:
a and b depends on the customer's order and varies.
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I was curious about the ratio of x0/b: if it is really small then, as has been mentioned, you can simply multiply x0 by the perimeter of the ellipse to get the area in question.
 
This topic seemed like a nice way to see how much Maple I have forgotten. So I have attached a pdf of a Maple worksheet where I took a = 2, b = 1 and k = 0.2. It could be used as a check of everyone's calculations, including mine. I didn't assume the inside curve was an ellipse.
 

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LCKurtz said:
This topic seemed like a nice way to see how much Maple I have forgotten. So I have attached a pdf of a Maple worksheet where I took a = 2, b = 1 and k = 0.2. It could be used as a check of everyone's calculations, including mine. I didn't assume the inside curve was an ellipse.
Click to expand...
1.812025940 your method gives this answer
1.812025938 and my code of post#20 agrees to lots of decimal places (shown in bold). This number was obtained after bumping up "n", the number of strips in the numerical integration.
 
That's pretty cool. If anybody cares, I have cleaned up the worksheet a bit which is attached.
 

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blamocur said:
I was curious about the ratio of x0/b: if it is really small then, as has been mentioned, you can simply multiply x0 by the perimeter of the ellipse to get the area in question.
Click to expand...
Aside from intellectual curiosity, this I think is much ado about nothing!!
Assuming we would have a max error of 5%, by assuming the inner curve is an ellipse (with axes of 'a - l' and 'b - l'), l propose that in a practical situation only the upper-bound of error will be 'somewhat' significant.
 
Subhotosh Khan said:
Assuming we would have a max error of 5%, by assuming the inner curve is an ellipse (with axes of 'a - l' and 'b - l'), l propose that in a practical situation only the upper-bound of error will be 'somewhat' significant.
Click to expand...
I agree that real world situations often don't require much accuracy. This must be true for painting, where I'm sure that other factors like the choice of paintbrush, painter, quality of finish etc would also cause changes of the amount of paint used.

But, it should be noted that the difference between the exact calculation and ellipse approximation can become greater than 5% if a/b > 2.45 (also depending on x0, and worst case here is 0.59b). And, the approximation becomes worse as the ellipse is stretched more. But it never seems to get worse than a 22% error (for VERY stretched ellipses).
 
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