# Computing mass with a denstiy function

1. Nov 11, 2009

### MasterWu77

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$$\leq$$ t $$\leq$$ $$\pi$$/2
and density function $$\rho$$(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. Nov 12, 2009

### HallsofIvy

Staff Emeritus
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
$$\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dx}\right)^2}dt$$.

However, here, you should be able to see that the density at point (x,y) is $cos^2(t)+ sin^2(t)= 1$. (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.