Tricky Integration Question

swal

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Jan 13, 2022
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5
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
 
Why is there a 2y in the denominator? You have 3 variables k,x,y and 1 equation. Please post the full original question.
 
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.
 
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:
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.
 
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.
 
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
 
I've always wondered why some posters insist on given the entire problem over a few posts.
 
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