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Getting the "Arc Length Function"
I have two problems scanned, one is an in class example and one is from the homework. The book uses the standard arc length of a curve equation to get the answers. Later in the same chapter they introduce the Arc Length Function, using 's' for the parameter.
My professor instructed us to use the normal equation but also to find s(t). The in-class example was really easy, as we only had to integrate a constant. In the HW example, when I setup the integral I end up having to do a substitution, when I already did a substitution going from r(t) to r(u). My understanding is that going from r(t) to r(u) is not a real substitution, but merely a change of dummy variables. Anyways I tried it on the HW problem and once I get my s(t), I get zero for my length if I evaluate it over the given interval.
I'm afraid I didn't get a clear explanation on how to find the arc length function and the book isn't much help either. Thank you!
The one I'm having issues is labeled #3 Sec 12.5, where I left it unevaluated.
Homework Statement
I have two problems scanned, one is an in class example and one is from the homework. The book uses the standard arc length of a curve equation to get the answers. Later in the same chapter they introduce the Arc Length Function, using 's' for the parameter.
My professor instructed us to use the normal equation but also to find s(t). The in-class example was really easy, as we only had to integrate a constant. In the HW example, when I setup the integral I end up having to do a substitution, when I already did a substitution going from r(t) to r(u). My understanding is that going from r(t) to r(u) is not a real substitution, but merely a change of dummy variables. Anyways I tried it on the HW problem and once I get my s(t), I get zero for my length if I evaluate it over the given interval.
I'm afraid I didn't get a clear explanation on how to find the arc length function and the book isn't much help either. Thank you!
Homework Equations
The Attempt at a Solution
The one I'm having issues is labeled #3 Sec 12.5, where I left it unevaluated.
Attachments
![Handwritten derivation of an arc length function for r(t) = t^3 i + t^2 j on [0, 2], with substitution 4 + 9u^2 = r and result s(t) = (1/18)[(4 + 9t^2)^(3/2) - 80 sqrt(10)].](/data/attachments/21/21688-19c30b34743bf7d1e2a71a4577e194be.jpg?hash=GcMLNHQ799)
![Handwritten notes on lined paper: #12 Sec 12.5 In class example, computing the arc length of r(t) = <2 sin t, 5t, 2 cos t> over [0, pi], with r'(t) and the length integral leading to s(t) = sqrt(29)(t - a) and s(pi) = sqrt(29) pi.](/data/attachments/21/21689-2ed1c18f3460a537508ceb4d1bc5de3c.jpg?hash=LtHBjzRgpT)
