This is my first task in this subject and I'm a bit stuck.
A thin wire is shaped with $f(x) = x^2$ where $2 \le x \le 3$.
Find the mass of the wire when density $\rho(x)=x$.
So I've done some reading and it seems like I need to use the formula for arclength. I'm not sure why, so my question is
Why should I use the formula to find bowlength? How do I solve these kind of problems?
Thank you.
Best Answer
Hint (use arclength): The mass is the integral $$ m = \int dm = \int \rho\, dl = \int_{x_1}^{x_2} \rho(x) \sqrt{1+f'(x)} $$ where $\rho$ is the density (a small piece of length $dl$ has mass $\rho \,dl$)