Arc length calculator calculus12/14/2023 ![]() We summarize these findings in the following theorem. So, eight times 26 is going to be 160 plus eight times six So, actually, let's justįigure that out, just for fun. And then we're gonna have 27 minus one inside, I guess you could This is going to be equal toĢ/3 times 4/9 is equal to 8/27. Actually, let's just factor out the 2/3, that makes it easier. Root of nine is three, to the third power is 27. To 4/9 times 2/3 times 9 to the 3/2 minus 2/3 times one to the 3/2. That at u equals nine and u is equal to one. It's going to be u to theģ/2, and then we divide by 3/2, which is the same Times the anti-derivative of the square root of u, Plugging our radius of 3 into the formula, we get C 6 meters or approximately 18.8495559 m. The circumference can be found by the formula C d when we know the diameter and C 2r when we know the radius, as we do here. Now we just need to find that circumference. Theorem of calculus here, to evaluate this definite integral. So, our arc length will be one fifth of the total circumference. And we know how to apply the fundamental, or, I guess, the second fundamental And I'm just gonna take theĤ/9 and stick it out here. Square root of u instead ofĭ x, we have times 4/9 d u. The definite integralįrom u is equal to one, to u is equal to 9- I'm gonna make it veryĮxplicit that I'm dealing with u now- of the square root of u. ![]() It clear that this is what is equal to this. And when x is equal to 32/9- and this is why that number was picked- what's u going to be equal to? 32/9 times 9/4 is gonna be 32/4, which is going to be eight plus one. 9/4 times zero is just zero, so u is going to be equal to one. I'm just gonna multiplyīoth sides times 4/9. Or, we could say d u is equal to 9/4 d x. So, if I say u is equal to one plus 9/4 x, then we know. I can kind of engineer that if I want, but instead, I'm just going toĭo straight up u-substitution. And you might be able toĮven do this in your head, essentially, do the u-substitution: say I have one plus 9/4 x. Have a definite integral that we know how to The formula for finding the arc length between two points is 1+(dy/dx)2 dx If we want to find the arc length of sin(x) from 0 to 1, then we just plug those. So this is f-prime of x į-prime of x squared is going to be this quantity squared. So we've done a lot ofĮngineering of this problem to make the numbers work out well, but let's just go through it. It's fairly straightforward to find the anti-derivative. Quite well when we put it under the radical, and And we picked this particular function because it simplifies Then f-prime of x is going to be 3/2 x to the 1/2. Now, what's the derivative? If f of x is x to the 3/2, To be the definite integral from zero to 32/9 of the ![]() It in general terms first, so that you can kinda see theįormula and then how we apply it. The arc length is going to beĮqual to the definite integral from zero to 32/9 of the square root. So let's just apply the arc length formula that we got kind of a conceptual proof for in the previous video. I'm working through it, you feel inspired, alwaysįeel free to pause the video and continue working with it. ![]() And I encourage you to pause the video and try this out on your own. BUT SOMETIMES THERE ARE SOME PROBLEMS, calculus has been beating me senseless but this coupled with YouTube videos has helped me out tremendously. Length right over here, this thing that I have depicted in yellow. This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. It's gonna be a little bit past three and 1/2, so it's Work out very well- to x is equal to 32/9. To when x is equal to- and I'm gonna pick a strange number here, and I picked this strange number 'cause it makes the numbers The arc length of this curve, from when x equals zero Over here, we have the graph of the function y is equal
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