calculating first kind surface integral

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I would like help calculating the next integral:

$\iint_Szds$ when $S=${$x=u\cos t$ , $y=u\sin t$ , $z=t$ , $0\le u\le1$ , $0\le t\le2\pi$}

Here is what I did:

$|r'_u\times r'_t|=\sqrt{1+u^2}$

so the integral is:

$\int_0^{2\pi}\int_0^1t\sqrt{1+u^2}\,\mathrm{d}u\,\mathrm{d}t$

But that does not seems right.

What am I doing wrong?