Thermal Physics question regarding the thermal expansion coefficients

[Stolbik]

Homework Statement

I am working on this ahead of my fall class and don't actually want the answer...
just pointers to help me understand something.. Thanks guys! :)
I am really rusty with my general physics and calculus knowledge =(

The original question asks me to prove that, for a solid, the linear thermal expansion coefficients (in x, y, z directions) add together to give the thermal expansion coefficient as such:

B=a_x+a_y+a_z

where B is the thermal expansion coefficient
B=(deltaV/V)/deltaT
deltaV= change in volume
V=volume
deltaT=change in temperature in Kelvin

and a is the linear thermal expansion coefficient
a=(deltaL/L)/deltaT
deltaL= change in length
deltaL=length

So here are my questions:

Shouldn't the equation be B=a_x*a_y*a_z instead? for a solid like a cube you have to multiply the lengths to get the volume... Why isn't it the same here?

Also is there a way to get the volume from the lengths of an object with calculus? I don't remember =( Just remember you can get the area under a curve from doing the integral or the volume of an object made by a curve somehow too...

Homework Equations

B=a_x+a_y+a_z

where B is the thermal expansion coefficient
B=(deltaV/V)/deltaT
deltaV= change in volume
V=volume
deltaT=change in temperature in Kelvin

and a is the linear thermal expansion coefficient
a=(deltaL/L)/deltaT
deltaL= change in length
deltaL=length

The Attempt at a Solution

uh.. well my attempt so far has been to understand the question. I tried to define the Volume as Lx+Ly+Lz but then got stuck. Please don't give me the answer though! I got 3 months to work this out :)

 

[Andrew Mason]

Shouldn't the equation be B=a_x*a_y*a_z instead? for a solid like a cube you have to multiply the lengths to get the volume... Why isn't it the same here?

You are trying to find the relationship between the change in volume and change in temperature.

You are given the relationship between change in length in the x, y and z directions. From that you can work out what an infinitesimal change of dx, dy and dz does to the change in volume (ie for an infinitesimal change in temperature dT):

V + dV = (x + dx)(y+dy)(z+dz) = (xy + xdy + dxy + dxdy)(z+dz)

Work that out and ignore the second and third order infinitesimal terms.

AM