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! =/

[Klaynos]

the p will probably be rho ([math]\rho[/math])

 

You need to do a volume integral, a dxdydz (in cartisians) of the density function between the limits:

 

x: 0->r_x

y: 0->r_y

z: 0->r_z

[cosine]

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

 

your p is very strange as it is the sum of a vector and a scalar value. i j and k are vectors. how they work is like this:

 

you could write the point (1, 2, 3) as 1i + 2j + 3k

 

also this p is weird because density is a scalar value, not a vector value, so there should be no i's j's or k's that don't cancel out in its formula.

 

when you have this figured out the straight forward way to look at this would be to use cylindrical co-ordinates. the cross-section of the object at any given z is a circle of radius 2. thats because cos t i + sin t j is the parametrization of a circle in a single variable.

 

maybe when you come back with more details on p we can help you out more

[iNow]

your p is very strange

 

<...>

 

maybe when you come back with more details on p we can help you out more

 

I don't know if I'll ever grow out of finding such immature interpretations taken so completely out of context so very funny.

[mooeypoo]

your p is very strange as it is the sum of a vector and a scalar value. i j and k are vectors. how they work is like this:

 

you could write the point (1, 2, 3) as 1i + 2j + 3k

 

also this p is weird because density is a scalar value, not a vector value, so there should be no i's j's or k's that don't cancel out in its formula.

 

when you have this figured out the straight forward way to look at this would be to use cylindrical co-ordinates. the cross-section of the object at any given z is a circle of radius 2. thats because cos t i + sin t j is the parametrization of a circle in a single variable.

 

maybe when you come back with more details on p we can help you out more

Density (rho) can be a function of volume, too, not necessarily scalar. For that matter, if the object is denser in the center and gets less dense as it moves away from the center, then the rho is not scalar, it's a function of space (x,y,z).

[Mr Skeptic]

I don't know if I'll ever grow out of finding such immature interpretations taken so completely out of context so very funny.

 

You didn't seem to think it was all that funny that time I did it to you. That said, I do think it is funny.

[cosine]

Density (rho) can be a function of volume, too, not necessarily scalar. For that matter, if the object is denser in the center and gets less dense as it moves away from the center, then the rho is not scalar, it's a function of space (x,y,z).

Oh I see, k in the p equation is the 'really k' from later on! Sorry CalleighMay! Um... are you still working on this problem?