Find the total mass of a wire with a certain density ?

[CalleighMay]

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help!

I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesday he asks us to discuss the problems as a class, seeing which ones of us know our stuff =P

Basically, i want to ask you guys what you think about these problems as i do them along before i have my discussion. I really want to make a lasting impression on my professor by "knowing my stuff" -to show him i can do it! All's i need is a little help! Would you guys mind giving me some help?

We are using the textbook Calculus 8th edition by Larson, Hostetler and Edwards and the problems come from the book.

The problem is on pg 1075 in chapter 15.2 in the text, number 26. It reads:

Find the total mass of the wire with density p.
And it gives:
r(t)=2 cos ti + 2 sin tj + 3tk
and p(x,y,z)=k+z
(the p is a different looking p, most likely represents something else, something that sounds like roe maybe? lol. and k is really k below)
and: (k>0), 0<=t<=2pi

I looked at similar problems in the same section and came up with the following for this one:

r'(t)=2 cos ti =2 sin tj
but when finding II r'(t) II how do i do this with sin and cos? I know it's sqrt of each term squared, so would it be: sqrt( 2cos^2t - 2sin^2t ) ?

Then at this point, even if the above was correct, it's telling me to do:

integral from C to ? of p(x,y,x) dx and integral from C to ? of kz ds

Yeah I'm lost! :( Any further help would be greatly appreciated. Thanks guys! =/

 

[Defennder]

You're told to find the mass of the wire. The mass of a differential segment of wire ds is p(x,y,z)ds, where p is the mass per unit length of the wire. Think of how to relate this to a scalar line integral.

 

[CalleighMay]

i'm sorry to seem stupid but i don't know what that is... i looked at similar problems and cannot follow the work at all.

 

[Defennder]

You need to understand the concept of a scalar line integral before you can do this question. You know that the differential mass dm(x,y,z) of a wire segment ds is $$dm = p(x,y,z)ds$$. Now how do you find M? You integrate both sides of the equation.

The general technique for this problem is to find r(t) vector function, which you have, then find $$|\textbf{r}'(t)|$$ and then express p(x,y,z) as a scalar function of t. So the line integral now becomes $$\int_{t_1}^{t_2} p(t) \left|\frac{d\textbf{r}(t)}{dt}\right| dt$$. You have to determine the values of t1 and t2 which gives you the starting and endpoints of r(t).