Answer
$(x, y, z) = (\frac{3\pi (2 \pi^2 + 1)}{4\pi^2 + 3}, 0, 0)$
Work Step by Step
$x = t$
$y = \cos(t)$
$z = \sin(t)$
$0 \leq t \leq 2\pi$
$m = \int_c(x^2 + y^2 + z^2) ds$
$= \int_c^{2\pi}f(t)\sqrt {(f_x(x, y, z)^2 + f_y(x, y, z)^2 + f_z(x, y, z)^2)}dt$
$ = \int_0^{2\pi} (t^2 + 1) \sqrt {(1)^2 + (-\sin t(t))^2 + (\cos(t)^2)} dt$
$ = \int_0^{2\pi}(t^2 + 1)\sqrt {2}dt$
$ = \sqrt {2}(\frac{8}{3}\pi^3 + 2\pi)$
$x = \frac{1}{m}\int_c xp(x, y, z)ds$
$ = \frac{1}{\sqrt {2}(\frac{8}{3}\pi^3 + 2\pi)}\int_0^{2\pi} t(t^2 + 1)\sqrt {2}dt$
$ = \frac{1}{\sqrt {2}(\frac{8}{3}\pi^3 + 2\pi)}\int_0^{2\pi}\sqrt {2} (t^3 + t^2)dt$
$ = \frac{4\pi^4 + 2\pi^2}{\frac{8}{3}\pi^3 + 2\pi}$ $ = \frac{3\pi(2\pi^2 + 1)}{4\pi^2 + 3}$
$y = \frac{1}{m}\int_c yp(x, y, z)ds$
$ = \frac{3}{2\sqrt {2}\pi(4\pi^2 + 3)}\int_0^{2\pi}(\sqrt {2}\cos(t))(t^2 + 1)dt$
$ = 0$
$z = \frac{1}{m}\int_c zp(x, y, z)ds$
$ = \frac{3}{2\sqrt {2}\pi(4\pi^2 + 3)}\int_0^{2\pi}(\sqrt {2}\sin(t))(t^2 + 1)dt$
$ = 0$
Therefore: $(x, y, z) = (\frac{3\pi (2 \pi^2 + 1)}{4\pi^2 + 3}, 0, 0)$
