Answer
$$\approx 4.1073$$
Work Step by Step
The parameterization for the given surface can be written as:
$r=\lt r \cos \theta, r \sin \theta, \cos r^2 \gt$
and $|r_r \times r_{\theta}| dA=\sqrt {4r^4 \sin^2 r^2+r^2}=r \sqrt {4r^2 \sin^2 r^2+1}$
Now, $$A(S) =\iint_{D} |r_r \times r_{\theta}| dA=\iint_{D} r \sqrt {4r^2 \sin^2 r^2+1} d \theta dr \\=\int_0^1 \int_0^{2 \pi} r \sqrt {4r^2 \sin^2 r^2+1} d \theta dr \\=2\pi \int_0^1 r \sqrt {4r^2 \sin^2 r^2+1} dr$$
Thus, by using a calculator, we get
$$A(S) = 2\pi \int_0^1 r \sqrt {4r^2 \sin^2 r^2+1} dr \approx 4.1073$$
