Tutorial Exercise

Evaluate ${\displaystyle \underset{S}{\overset{\phantom{}}{}}}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x,y,z)dS$

$f(x,y,z)=\sqrt[\phantom{2}]{x^2+y^2+z^2}$

$S$:$z=\sqrt[\phantom{2}]{x^2+y^2},(x-1{)}^2+y^{2}1$

Step $1$

Find the surface integral using the formula

${}_{S}f(x,y,z)dS={}_{R}f(x,y,g(x,y))\sqrt[\phantom{2}]{1+[g_x(x,y){]}^2+[g_y(x,y){]}^2}dA$

Here, $g(x,y)=z=\sqrt[\phantom{2}]{x^2+y^2}.$ Find the partial derivatives $g_x(x,y)$ and $g_y(x,y).$

$g_x(x,y)=\frac{?}{\sqrt[\phantom{2}]{x^2+y^2}}$

$g_y(x,y)=\frac{?}{\sqrt[\phantom{2}]{x^2+y^2}}$