Struggling with LaTeX: Seeking Help With String Length & Mass

[mikeyrichster]

Im afraid my use of LaTeX code, sucks. My apologies to anyone friendly enough to help!

The problem:

A string takes a path shown by the equation below:

r(t)=(t,3t^2,6t^3) Where the RHS is a vertical vector (didnt know how to code this!)

and 0 <= t <= 1

The mass per unit length of the wire at a point (x, y, z) is given by
p(x,y,z) = xyz^2

Im having trouble answering these last two questions..

c) Find the entire length of the string?
d) Find the mass of the string?

Thanks for your help!

 

[eok20]

For c) you're being asked for the arc length. Do you know the expression that gives you arclength for a parameterized curve? It's not too bad to remember/derive: if you think of t as time, then the distance (not displacement) that the particle goes at time dt is the speed at the time, which is |r'(t)|, times dt. Therefore the total distance will be the integral from t = 0 to 1 of |r'(t)|dt.

 

[mikeyrichster]

Hi there, thanks for your help!

Is the expression you refer to the arc length formula?

you said "therefore the total distance will be the integral from t = 0 to 1 of |r'(t)|dt."
does that mean the integral from 0 to 1 of the absolutle vaule of r(t) differentiated?

How do you integrate vectors like this?

Thanks

 

[eok20]

By |r'(t)| I meant the norm (magnitude) of the vector r'(t). So if r(t) = (x(t), y(t), z(t)) then r'(t) = (x'(t), y'(t), z'(t)) so that |r'(t)| = sqrt(x'(t)^2 + y'(t)^2 + z'(t)^2).