Question

find the mass of a wire shaped like a semicircle if it's density is 2xy kg/dm3

Answer 1

....kg/dm i mean

Answer 2

does the question state what the radius of the semi-circle is, or where it is positioned in space?, since if I assume it to be on the origin then the mass is zero

Answer 3

It's the circle y^2+x^2=9,

Answer 4

\[M=\int\limits_{C} 2xy ds\]
1st parametrise the x and y functions
\[x=3 \cos(\theta)\]

\[y=3 \sin(\theta)\]
\[ds=\sqrt{f'(t)^2+g'(t)^2}\]
\[ds=\sqrt{9(\cos(\theta)^2+\sin(\theta)^2)}\]
\[ds=3d\theta\]
therefore
\[M=\int\limits_{0}^{\pi} 2\times 3\cos(\theta)\times3\sin(\theta)\times3d\theta\]
\[M=54\int\limits_{0}^{\pi} \sin(\theta)\cos(\theta)d\theta\]
using a substitution \[M=27\sin(\theta)^2|_0^\pi\]
Therefore
\[M=0\]

Answer 5

yes makes sence, x is negative between pi/2 and pi, thus the integral is an odd function of x and hence by symmetry we get 0. The density function is unrealistic

Answer 6

in a mathematical sense it is correct :P

Answer 7

lol, x^2y would make more sence. I take it you know ur vector calculus