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Numerade Educator

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Problem 46 Easy Difficulty

The curves with equations $ x^n + y^n = 1 $, $ n = 4, 6, 8 \cdots , $ are called fat circles. Graph the curves with $ n = 2, 4, 6, 8, $ and $ 10 $ to see why. Set up an integral for the length $ L_{2k} $ of the fat circle with $ n = 2k $. Without attempting to evaluate this integral, state the value of $ \lim_{k\to\infty} L_{2k} $.

Answer

$$\lim _{k \rightarrow \infty} L_{2 k}=8$$

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Video Transcript

So we're going to be looking at curse with the equations next to the end plus Y. To the hand equals one Where energy click of 468. So look at two and then we'll zoom in states are called fat circles. And then if we go to four six mhm. 10. And what we can expect is that as this gets larger, it's essentially going to end up giving us a square. So it's going to be the limit As K goes to infinity for the limit of two K. And as far as the integral goes, we wanna remember the arc length integral. So it's going to be the length is equal to the integral from alpha to beta or a to B. Of the square root of the X. Um B. T squared. And we can write this with in the former parametric equations. Or we can also do it in the form of polar equation if that might be the easiest if we know how you polar equations. Um But either way we'll do this with respect the D. T. Or we can configure this in a way that allows us to sulfur in terms of just X and Y. And that will give us our final answer for the arc length