Problem 22 Hard Difficulty

Find the length of the arc of the curve from point $ P $ to point $ Q $.

$ x^2 = (y - 4)^3 $ , $ P(1, 5) $ , $ Q(8, 8) $

Answer

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Video Transcript

Hey, it's clear. So when you're in here so we have X square is equal to why minus four kids, we have two points. He, uh, when comma five, two, meet comma, Ain't birth gonna solve for X? We get plus or minus square e of why minus four. Cute. We know that the points of positive X coordinates. So we're only then I used the positive, so e equals a lie minus four to the 3/2 policy. We're gonna find d X over D Y in the equals three house. Why? Minus four? No one. How Now we're gonna plug it into our equation. Park links equations from five to eight. Square root of one plus nine over four times. Why minus four day? Why? It's five and eight since we're finding it in respect to why we're integrating in respect to why who has becomes equal to from 5 to 8. Square of one plus nine over four. Why minus and nine. Do you? Why becomes a full too integral five square root of nine. Why minus 32 Do you know why I put a 1/2 here? Louis, integrate this. It becomes on the house times, 2/3 times won over nine times nine. Why minus 32 to the three halves, 5 to 8. This is around one over 27. I'm 80 square of 10 minus 13 square of 13 which is around seven point 6337