Problem 10 Medium Difficulty

Find the exact length of the curve.

$ 36y^2 = (x^2 - 4)^3 $ , $ 2 \le x \le 3 $ , $ y \ge 0 $

Answer

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Video Transcript

first will want to solve for y so immediately we can see that we'll need to divide by 36. Getting why squared equals one over 36 times X squared minus four. You then we can square root both sides to get why equals 16 times X squared minus for to the three seconds now Because we are given that why is greater than or equal to zero we will only need the positive of the square roots. We won't have to worry about the negative one. You can go to our next page here and well, you will need to take the derivative of why so we can see that 16 times. Three over two times X squared minus four to the 1/2 now and then times the derivative of X squared minus four, which is two X Now the twos cancel. In this case, three over six is going to get this 1/2. So why prime equals 1/2 Times X? Because we have to remember our X over here. Times X squared minus four to the 1/2. So going over to our next page, we have our derivative. Now we can plug this in to our equation. Um, we were given the interval two three so we can plug in to for a on three or b radical one plus 1/2 x times X squared minus four to the 1/2 squared D X. Now let's go back for a second. Um, we will square this. So this is 1/4 x squared times X squared, minus four and then squaring. This eliminates the, um, 1/2 power here. So if we go over here, we end up getting 1/4 extra the fourth minus four x squared. Now we'll go to our next page, and we can factor out 1/4 and then we can move it outside and reorder terms the bit. And now we have X the fourth minus X squared, plus her minus four x squared plus four d x. And we can see that this can be also written as a radical X squared minus two squared D X. And now, um oh, I've already written that here. Oh, no. Okay, pardon me. So in this part will simplify um, rather integrate. So x squared will be x cubed. 1/3 minus two x now um, still with our boundaries. Um, 223 Um, And then we'll come to the next page where I've already written this out for us. So this comes down to 1/2 9 minus six, minus eight over three, minus four. And then, uh, simplifying about more. We have nine minus six equals three minus eight over three. Um, you have plus four. Distributing this negative equals 1/2 seven minus eight over three. Um, we'll continue to simplify. So this way we can write 21 over three for seven blindness, So this will give us 13 over three times 1/2 which equals 13 over six, which is our final answer.