Tricky Integration Question

swal

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Jan 13, 2022
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Hello,

I'm having trouble solving the following equation; could someone please share their solution? They have given the arc length, but now I need to solve for k in the equation. Thanks! Screen Shot 2022-01-13 at 5.28.54 pm.png
 

swal

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Jan 13, 2022
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Why is there a 2y in the denominator? You have 3 variables k,x,y and 1 equation. Please post the full original question.
The function is a relation - I'm trying to find the length of the curve of this function:
Screen Shot 2022-01-14 at 2.33.18 pm.png
Where a is 2.95 and b is 2. I need the length of the curve between -2.95 and 2.95, given y is greater than or equal to zero.This can be rearranged as: Screen Shot 2022-01-14 at 2.34.16 pm.png
This can be implicitly differentiated in order to substitute into the formula for the length of an arc, but I can't do the integration.
 

BigBeachBananas

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This can be implicitly differentiated in order to substitute into the formula for the length of an arc, but I can't do the integration.
Take the square root?
[math]y=\pm \sqrt{\left(1-\frac{x^2}{a^2}\right)\frac{b^2}{1+kx}}[/math]
y is greater than or equal to zero
Take the positive only.
[math]y=\sqrt{\left(1-\frac{x^2}{a^2}\right)\frac{b^2}{1+kx}}[/math]First, I would simplify then find the derivative. But it does not look pretty...
 
Last edited:

Dr.Peterson

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Nov 12, 2017
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The function is a relation - I'm trying to find the length of the curve of this function:
View attachment 30672
Where a is 2.95 and b is 2. I need the length of the curve between -2.95 and 2.95, given y is greater than or equal to zero.This can be rearranged as: View attachment 30673
This can be implicitly differentiated in order to substitute into the formula for the length of an arc, but I can't do the integration.
The integrand must be a function of x; implicit differentiation doesn't initially produce that, because you still have y in the derivative. You need to replace y with its expression in terms of x (BBB's radical) before integrating. (Assuming the rest of your work is correct, which I haven't looked at.)

Not all functions can be integrated exactly, or with a reasonable amount of work! This may well be one of the many that would need numerical integration.
 

BigBeachBananas

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I feel like you haven't posted the full question and its instruction. For the integral to converge, [imath]-0.3 \le k \le 0.3[/imath]. Notice for [imath]k=0[/imath], it's an ordinary ellipse. Was this info given? Are you expect to integrate by hand? You haven't given us the full context of the question.
 

swal

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Jan 13, 2022
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ok, so I've done that. Can anyone integrate this with respect to x in order to find k?
Screen Shot 2022-01-15 at 10.41.15 am.png
 

Jomo

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I've always wondered why some posters insist on given the entire problem over a few posts.
 
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