Question

Find the area of the paraboloid x2 + y2 = z inside the cylinder x2 + y2 = 9.

Answer 1

$z_x = 2x$

$z_y=2y$

The surface area over the region R defined by $x^2+y^2 = 9=3^2$ is

$S = \int \int_R \sqrt{(z_x)^2+(z_y)^2+1}dxdy$

= $\int \int_R \sqrt{4x^2+4y^2+1}dxdy$

Then to polar coordinates

$S = \int_{0}^{2\pi} \int_0^3(4r^2+1)^{1/2}rdrd\theta$

$= \dfrac{1}{12} \int_{0}^{2\pi} (4r^2+1)^{3/2}|_0^3d\theta$

$= \dfrac{37\sqrt{37} -1}{12} \int_0^{2\pi} d\theta$

$=\dfrac{37\sqrt{37} -1}{12} 2\pi$

$= \pi \dfrac{37\sqrt{37}-1}{6}$