The question asks to find the length of the curve r(t) = <2+3t, 1-4t, -4+3t> from (5,-3,-3) to (20,-23,14). I took the derivative of r(t) and then found its magnitude. I then took the integral of ||r'(t)|| and got sqrt(34)t. What point do i plug in for t? In the back of the packet are the answers and it says the answer is 5sqrt(34). Do you always just plug in the first point given?
Thank you,
*************************
Well, you know you need to compute an integral, but it has to be a definite integral ∫_a^b ||r'(t)|| dt, and you have discovered that real (and only) trick that is required is to figure out the limits of integration a and b. You can figure this out from the clause "from (5,-3,-3) to (20,-23,14)"; this means that at the start the curve is at (5,-3,-3), which says that a is the solution of
<5,-3,-3>= <2+3t, 1-4t, -4+3t>
which gives you that t=1. Then at the end the curve is at the point (20, -23, 14), which gives the equation <20,-23,14>= <2+3t, 1-4t, -4+3t> has the solution t=6, so your integral is from 1 to 6.
