Answer
$\approx 3.8857$
Work Step by Step
Write the parameterization representation for the given surface as: $r=\lt r \cos \theta, r \sin \theta, \cos r^2 \gt$
Since, $Surface \ Area ; A(S)=\iint_{D} |r_r \times r_{\theta}| dA$
and $\iint_{D} dA$ is the area of the region $D$
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, $Surface \ Area ; A(S) = \iint_{D} |r_r \times r_{\theta}| dA \\=\iint_{D} r \sqrt {4r^2 \sin^2 r^2+1} \ d \theta \ dr \\=\int_0^1 \int_0^{2 \pi} r \sqrt {4r^2 \sin^2 r^2+1} d \theta dr\\=2\pi \int_0^1 r \sqrt {4r^2 \sin^2 r^2+1} \ dr$
Now, we will use calculator, so we have:
$A(S) = 2\pi \int_0^1 r \sqrt {4r^2 \sin^2 r^2+1} dr \approx 3.8857$
