# Thread: arc Length of a spiral

1. ## arc Length of a spiral

HI,

I need to find the length of a spiral given by r(t) = pi*(t*cos(pi*t)i - t*sin(pi*t)j +t k) , from t=0 to t=3
I have been given the formula

integral (sqrt(a^2 + b^2*t^2)dt = 1/2*t*sqrt(a^2 + b^2*t^2) + (a^2/2b)*ln(b*t + sqrt(a^2 + b^2*t^2).

I proceeded to find dot product of tangent vectors

r'(t) . r'(t) = (pi*cos(pi*t) - pi^2*tsin(pi*t))^2 + pi^2(pi*t*cos(pi*t) + sin(pi*t)) +pi^2

However, now i have come unstuck, where do i go from here?

I was thinking about letting a^2 = (pi*cos(pi*t) - pi^2*tsin(pi*t))^2,

b^2= (pi*t*cos(pi*t) + sin(pi*t)) +1, t^2=pi^2

then substituting into the formula and solving t=3 and t=0

But i really have no idea where to get the parameters a , b , t from to use in the formula

Any help would be greatly appreciated.

2. Originally Posted by olski1
HI,

I need to find the length of a spiral given by r(t) = pi*(t*cos(pi*t)i - t*sin(pi*t)j +t k) , from t=0 to t=3
I have been given the formula

integral (sqrt(a^2 + b^2*t^2)dt = 1/2*t*sqrt(a^2 + b^2*t^2) + (a^2/2b)*ln(b*t + sqrt(a^2 + b^2*t^2).

I proceeded to find dot product of tangent vectors

r'(t) . r'(t) = (pi*cos(pi*t) - pi^2*tsin(pi*t))^2 + pi^2(pi*t*cos(pi*t) + sin(pi*t)) +pi^2

However, now i have come unstuck, where do i go from here?

I was thinking about letting a^2 = (pi*cos(pi*t) - pi^2*tsin(pi*t))^2,

b^2= (pi*t*cos(pi*t) + sin(pi*t)) +1, t^2=pi^2

then substituting into the formula and solving t=3 and t=0

But i really have no idea where to get the parameters a , b , t from to use in the formula

Any help would be greatly appreciated.
I think you need to check your derivative. I get that

$\mathbf{r}'(t)=\pi[(\cos(\pi t)-t\pi \sin(\pi t))\mathbf{i}+(-\sin(\pi t)-t\pi \cos(\pi t))\mathbf{j}+\mathbf{k}]$

After dotting this with itself and simplifying I get

$\mathbf{r}'(t) \cdot \mathbf{r}'(t)=\pi^2(\pi^2t^2+2)$

So the integral should be

$\displaystyle \pi \int_{0}^{3}\sqrt{\pi^2t^2+2}$

Now use the substitution $\displaystyle t= \frac{\sqrt{2}}{\pi}\sinh(x)$