Complex Integration for polar coordinates

Question

I need to find $\int_{\gamma} |z|^2dz$ for $\gamma[0,1]\rightarrow\mathbb{C}$ with $\gamma(t)=r(\cos(2\pi t)+i\sin(2 \pi t)) $. I will use the formula: $$ \int_{\gamma}f(z)dz=\int_0^1f(\gamma(t))\gamma'(t)dt $$ and using that for $z=x+iy$ it is $|z|=\sqrt{x^2+y^2}$ we get $|z|^2=x^2+y^2$. First I tought about using Eulers formula for $r(\cos(2\pi t)+i\sin(2 \pi t))$ but it seems that this can be done much easier, tough not quite sure because it seems too easy. So: $$ |r\cos(2\pi t)+r i\sin(2 \pi t))|^2=r^2\cos^2(2\pi t)+r^2\sin^2(2\pi t)=r^2(\cos^2(2\pi t)+\sin^2(2\pi t)) = r^2 $$ since $\cos^2(2\pi t)+\sin^2(2\pi t)=1$. So we get $$ \int_{\gamma}f(z)dz=\int_0^1f(\gamma(t))\gamma'(t)dt = \int_0^1r^2\gamma'(t)dt=r^2\int_0^1\gamma'(t)dt = r^2[\gamma(t)]_0^1 = r^2(r(\cos(2\pi)+i\sin(2 \pi))-r(\cos(0)+i\sin(0)))=r^2(r(1+0)-r(1+0))=r^2(r-r)= 0 $$ Does this make sense and if not, where ist the mistake? Thank you guys in advanc!