# Thread: calc 3- line integral

1. ## calc 3- line integral

find the mass of a wire in the shape of the helix traced by (cost, sint, t/pi)
from pi to 3 pi if its density at each point is proportional to the distance from the point to the xy-plane

so i have mass= the integral from pi to 3pi of
cost + sint + t/pi dL

but i'm not sure what dL is..i saw it in an example in the book.
is dL the derivative of what i have above? or is it the magnitude? or do i ignore it?
after i clarify this, i think i'll be able to hopefully solve it..

2. $dL$ means that the integral is with respect to arc length. Imagine if the wire had a uniform mass density - then its mass would clearly be equal to its length times the mass density.

You probably know that the length of a parametrized curve $\phi$ between $\phi(a)$ and $\phi(b)$ is

$L=\int_a^b|\phi'(s)|ds$

so $dL = |\phi'(s)|ds$. Therefore the integral you are looking for is $\int_{\pi}^{3\pi}W(\phi(s))|\phi'(s)|ds$ where $W(\phi(s))$ is the mass density of the wire at the point $\phi(s)$. You are told that this is proportional to the distance from the point $\phi(s)$ to the $XY$ plane, i.e. $W(\phi(s))=k \times d(\phi(s), XY\mbox{ plane})$ for some constant $k$. Find the expression for that, substitute it into the integral and then evaluate it.

3. i'm sorry i'm sort of confused with all of your symbols. my final answer is:
4*sqrt(pi^2 +1)
is that right/close?

4. I didn't do the calculation! Please post your work, and I'll tell you if it's good.

5. f(x)= cost+sint+t/pi
dL= -sint+cost+1/pi then i took the magnitude of this to get
dL= (sqrt.(pi^2+1))/(pi)

so i took the integral from pi to 3pi of f(x)*dL
and got

(sint-cost+(1/2pi)*t^2)*(sqrt(pi^2 +1))/(pi))* t

and evaluated from 3pi to pi
and got

6. Well the function isn't $f(t)=\cos t+\sin t+\frac{t}{\pi}$, it's $f(t) = (\cos t, \sin t, \frac{t}{\pi})$.
Moreover, what did you do of the fact that the mass density at the point $f(t)$ is proportional to the distance between $f(t)$ and the $XY$ plane? If you do not take that into account in your solution then your solution is certainly wrong.