Skip to main content\(\require{physics}\require{upgreek}\everymath{\displaystyle}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\inv}{^{-1}}
\newcommand{\DS}{\displaystyle}
\newcommand{\eps}{\varepsilon}
\newcommand{\tsub}[1]{_{\mathrm{#1}}}
\newcommand{\ee}{\mathrm{e}}
\newcommand{\ii}{\mathrm{i}}
\newcommand{\limit}[1]{\lim\limits_{#1}}
\newcommand{\resid}[1]{\underset{#1}{\Res}}
\DeclareMathOperator{\sinc}{sinc}
\DeclareMathOperator{\sgn}{sgn}
\newcommand{\pii}{\pi}
\DeclareMathOperator{\Prob}{P}
\DeclareMathOperator{\EV}{E}
\DeclareMathOperator{\Var}{Var}
\newcommand{\bv}[1]{\boldsymbol{#1}}
\newcommand{\uv}[1]{\hat{\bv{#1}}}
\newcommand{\cl}[1]{\mathcal{#1}}
\newcommand{\bb}[1]{\mathbb{#1}}
\DeclareMathOperator{\Cis}{cis}
\DeclareMathOperator{\RE}{Re}
\DeclareMathOperator{\IM}{Im}
\newcommand{\xd}{\mathbf{d}}
\newcommand{\seq}[3]{{#1}_{#2},\cdots,{#1}_{#3}}
\newcommand{\psup}[1]{^{(#1)}}
\newcommand{\hypext}{{}^*}
\DeclareMathOperator{\st}{st}
\newcommand{\set}[1]{\left\{#1\right\}}
\DeclareMathOperator{\Sin}{Sin}
\DeclareMathOperator{\Cos}{Cos}
\DeclareMathOperator{\Tan}{Tan}
\DeclareMathOperator{\Sec}{Sec}
\DeclareMathOperator{\Csc}{Csc}
\DeclareMathOperator{\Cot}{Cot}
\DeclareMathOperator{\Log}{Log}
\DeclareMathOperator{\Arg}{Arg}
\DeclareMathOperator{\Ln}{Ln}
\DeclareMathOperator{\Grad}{grad}
\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\Curl}{curl}
\newcommand{\rd}{\textstyle\mathop{}\!\mathrm{d}^{\!\!\!-}\hspace{-0.0555 em}}
\newcommand{\rpd}{\textstyle\mathop{}\!\partial^{\hspace{-0.5 em}-}\hspace{-0.1666 em}}
\newcommand{\rdv}[2]{\frac{\rd{#1}}{\rd{#2}}}
\newcommand{\rpdv}[2]{\frac{\rpd{#1}}{\rpd{#2}}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 15.3 Surface area
Objectives
-
Compute the area of a surface
\(z = f(x,y)\) by integrating a surface-area element over its domain.
-
Relate the surface-area formula to the structure of the arc length formula in one dimension.
-
Interpret the integrand as the magnitude of a cross product formed from tangent vectors on the surface.
-
Area of the surface \(z=f(x,y)\) over the region \(D\text{:}\)
\begin{equation*}
\iint_{S}\dd{S}=\iint_{D}\sqrt{\qty(\pdv{f}{x})^{2}+\qty(\pdv{f}{y})^{2}+1}\dd{A}
\end{equation*}
Notice similarity with arc length formula.
-
Derivation has to do with surface area element:
\begin{align*}
\bv{a} \amp = \qty(\dd{x},0,\pdv{f}{x}\dd{x}) \\
\bv{b} \amp = \qty(0,\dd{y},\pdv{f}{y}\dd{y}) \\
\dd{S} \amp = \norm{\bv{a}\times\bv{b}} = \sqrt{\qty(\pdv{f}{x})^{2}+\qty(\pdv{f}{y})^{2}+1}\dd{A}
\end{align*}
