Computing mass with a denstiy function

[MasterWu77]

Homework Statement

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

Homework Equations

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

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

 

[HallsofIvy]

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.