Multivariable Calculus Length of Curve

Question

I have to find the length of a curve C which is parametrized by $x(t)=\dfrac{e^t+e^{-t}}{2},$ $y(t)=\cos(t)$ and $z(t)=\sin(t)$ where $t$ goes from -1 to 5. I believe this involves simply finding the length of the derivatives, but I am not sure what to multiple that with and something just seems a little off with my answer.

Answer 1

The arc-length formula is simply:

$$L = \displaystyle\int_a^b ||\vec{x}'(t)|| dt = \int_a^b \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} dt$$

So for this problem, you need to compute the integral:

$$L = \int_{-1}^5 \sqrt{\left(\dfrac{e^t - e^{-t}}{2}\right)^2 + (-\sin(t))^2 + \cos(t)^2} dt$$

Note that the integrand can be simplified quite a bit.

Answer 2

$$\int_{-1}^5ds=\int_{-1}^5\sqrt{sinh^2(t)+1}dt=\int_{-1}^5coshtdt=sinh(5)-sinh(1)$$