## Calculus 8th Edition

a) The surface integral of a scalar function $f(x,y,z)$ over the surface $S$ can be defined as: $\iint_Sf(x,y,z)dS$ b) A parametric surface let us say $S$ is known to be a surface in $R^3$ with two parameters $(m,n)$ represented as: $r(m,n)=p(m,n)i+q(m,n)j+r(m,n)k$ Here, $p,q,r$ are scalar functions. Area of a surface parametric surface $S$ can be defined as: $\iint_Sf(x,y,z)dS=\iint_Df(r(m,n))|r_m \times n_v|$ dA where $(m,n) \in D$ c) Area of the surface $S$ of equation $z=g(x,y)$ can be calculated as: $\iint_Sf(x,y,z)dS=\iint_Sf(x,y,g(x,y))\sqrt {1+(g_x)^2+(g_y)^2}dA=\iint_Sf(x,y,g(x,y))\sqrt {1+(\dfrac{\partial z}{\partial x})^2+(\dfrac{\partial z}{\partial y})^2}dA$ d) Mass of the thin sheet has the shape of a surface $S$ can be expressed as: $m=\iint_S \rho(x,y,z) dS$ Center of mass of the sheet can be defined as:$(\bar{x},\bar{y},\bar{z})$ Here, $\bar{x}=\frac{1}{m}\iint_Sx \rho(x,y,z) dS$; $\bar{y}=\frac{1}{m}\iint_Sy \rho(x,y,z) dS$; $\bar{z}=\frac{1}{m}\iint_Sz \rho(x,y,z) dS$