The discussion focuses on computing the arclength function s(t) for the curve defined by r(t) = (e^−2t cos(3t), e^−2t sin(3t), e^−2t). The derivative r'(t) is calculated as r'(t) = <-2e^-2t cos(3t) + e^-2t * -3sin(3t), -2e^-2t sin(3t) + e^-2t cos(3t), -2e^-2t>. To find the arclength, the integral s(t) = ∫_0^t ||r'(u)|| du is established, emphasizing the need for clarity in notation and the use of a dummy variable for integration.
PREREQUISITESStudents studying calculus, particularly those focusing on parametric equations and arclength calculations, as well as educators seeking to clarify these concepts for their students.
A minor point: the e^(-2t)*cos(3t) term in your second component is missing a factor of 3. Also, be sure to use plenty of parentheses to remove any ambiguity in the future!withthemotive said:r'(t) = <-2e^-2t*cos(3t) + e^-2t*-3sin(3t), -2e^-2t*sin(3t) + e^-2t*cos(3t), -2e^-2t>
That's right, but your derivative will need to be with respect to a dummy variable, say u:Then what, do I find the length of that derivative?
Then take the integral of 0 to t?