Solving for the mass using Parametric Equations (Density Forumula)

[EpiGen]

Hey,

I have a calculus 3 test coming up that involves calculating the mass of an object using a density function. I know how to use the double integral form for x-y equations, but the practice test my professor gave me has a question using parametric equations and I don't know how to calculate the answer.

Find the mass of a spring in the shape of a helix defined by r(t) = <2cost,t,2sint>, for 0<=t<=5*pi, density function p(x,y,z)=y.



I know the mass is calculated using Int(Int(p(x,y)dx)dy) for a 2 variable system but don't know how to incorporate the t or z variables.

 

[HallsofIvy]

You are mistaken in thinking that "the mass is calculated using Int(Int(p(x,y)dx)dy) for a 2 variable system". That is correct for a two-dimensional situation- a surface. What you have is a one dimensional figure in three dimensions. The mass is given by $\int \rho(x) d\sigma$ where $d\sigma$ is the "differential of arc length. If the figure is given by x= f(t), y= g(t), z= h(t), as you have here, then
$$d\sigma= \sqrt{\left(\frac{df}{dt}\right)^2+ \left(\frac{dg}{dt}\right)^2+ \left(\frac{dh}{dt}\right)^2}dt$$.