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Computing mass with a denstiy function

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Compute the total mass of a wire bent in a quarter circle with parametric equations: x=cos(t), y=sin(t), 0[tex]\leq[/tex] t [tex]\leq[/tex] [tex]\pi[/tex]/2
    and density function [tex]\rho[/tex](x,y) = x^2+y^2


    2. Relevant equations

    not exactly too sure which equations if any i need to use. maybe the jacobian

    3. The attempt at a solution

    i simply substituted the x and y into the density function to get

    (6cost)^2 + (6sint)^2 and took the integral of that with the bounds of integration from 0 to pi/2. the answer i am getting is 56.549 and is wrong and i'm not sure if there's an extra step i need to do
     
  2. jcsd
  3. Nov 12, 2009 #2

    HallsofIvy

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    Staff Emeritus
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    You want, of course, to integrate the density function along the length of the given curve. The differential of length, when the curve is given by parametric equations, x= x(t), y= y(t), is
    [tex]\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dx}\right)^2}dt[/tex].

    However, here, you should be able to see that the density at point (x,y) is [itex]cos^2(t)+ sin^2(t)= 1[/itex]. (I have no idea where you got the "6" in your formula). Since that density is constant around the arc, the mass is just 1 times the length of the curve. And that is simply 1/4 the circumference of a circle of radius 1.
     
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