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 verticle 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).

 

[blue_raver22]

I understand that learning a new coding language can be challenging and it is completely normal to struggle with it at first. My advice would be to practice and seek help from online resources or colleagues who are familiar with LaTeX. Additionally, there are many LaTeX forums and communities where you can ask for assistance with specific coding problems, such as the one you have mentioned.

In regards to the problem at hand, it seems like you are on the right track with the given equations. To find the length of the string, you can use the formula for arc length in three dimensions, which is given by L = ∫√(dx^2 + dy^2 + dz^2). You can plug in the values for x(t), y(t), and z(t) from the given equation and integrate from 0 to 1 to find the total length of the string.

To find the mass of the string, you can use the formula for mass per unit length, which is given by m = ∫p(x,y,z)ds, where ds is the infinitesimal length element. Again, you can plug in the values for x(t), y(t), and z(t) from the given equation and integrate from 0 to 1 to find the total mass of the string.

I hope this helps and don't be afraid to ask for help when needed. Keep practicing and you will improve your LaTeX skills in no time!