$\sqrt{h_x^2\+h_y^2\+h_z^2}$

while the notation $\left|h_z\right|$ stands for the absolute value of the partial with respect to $z$.

Tool

Comments

Int and/or int commands at top level

SurfaceArea command in the Student MultivariateCalculus package

Surface-area task template

SurfaceInt command in the Student VectorCalculus package

Table 6.3.3   Maple tools for surface integration

Top-level Int

$\mathrm{Int}\left(\mathrm{\lambda }\,\left[y\=\dots \,x\=\dots \right]\right)$ ⇒ ${\int }_{x_L}^{x_R}{\int }_{y\=y_B\left(x\right)}^{y\=y_T\left(x\right)}\mathrm{\lambda } \mathrm{dy} \mathrm{dx}$ 

$\mathrm{Int}\left(\mathrm{\lambda }\,\left[x\=\dots \,y\=\dots \right]\right)$ ⇒ ${\int }_{y_B}^{y_T}{\int }_{x\=x_L\left(y\right)}^{x\=x_R\left(y\right)}\mathrm{\lambda } \mathrm{dx} \mathrm{dy}$ 

MultiInt

$\mathrm{MultiInt}\left(\mathrm{\lambda }\,y\=\dots \,x\=\dots \right)$ ⇒ ${\int }_{x_L}^{x_R}{\int }_{y\=y_B\left(x\right)}^{y\=y_T\left(x\right)} \mathrm{\lambda } \mathrm{dy} \mathrm{dx}$ 

$\mathrm{MultiInt}\left(\mathrm{\lambda }\,x\=\dots \,y\=\dots \right)$ ⇒ ${\int }_{y_B}^{y_T}{\int }_{x\=x_L\left(y\right)}^{x\=x_R\left(y\right)} \mathrm{\lambda } \mathrm{dx} \mathrm{dy}$ 

SurfaceArea

$\mathrm{SurfaceArea}\left(f\,x\=\dots \,y\=\dots \right)$ ⇒ ${\int }_{x_L}^{x_R}{\int }_{y\=y_B\left(x\right)}^{y\=y_T\left(x\right)}\mathrm{\λ} \mathrm{dy} \mathrm{dx}$ 

$\mathrm{SurfaceArea}\left(f\,y\=\dots \,x\=\dots \right)$ ⇒ ${\int }_{y_B}^{y_T}{\int }_{x\=x_L\left(y\right)}^{x\=x_R\left(y\right)}\mathrm{\λ} \mathrm{dx} \mathrm{dy}$ 

SurfaceInt

$\mathrm{SurfaceInt}\left(g\left(x\,y\right)\,\left[x\,y\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,f\left(x\,y\right)⟩\,x\=\dots \,y\=\dots \right)\right)$ ⇒ ${\int }_{x_L}^{x_R}{\int }_{y\=y_B\left(x\right)}^{y\=y_T\left(x\right)}g\left(x\,y\right) \mathrm{\lambda } \mathrm{dy} \mathrm{dx}$

$\mathrm{SurfaceInt}\left(g\left(x\,y\right)\,\left[y\,x\,z\right]\=\mathrm{Surface}\left(⟨x\,y\,f\left(x\,y\right)⟩\,y\=\dots \,x\=\dots \right)\right)$ ⇒ ${\int }_{y_B}^{y_T}{\int }_{x\=x_L\left(y\right)}^{x\=x_R\left(y\right)}g\left(x\,y\right) \mathrm{\λ} \mathrm{dx} \mathrm{dy}$

int

(as modified in Vector Calculus)

$\mathrm{int}\left(\mathrm{\λ}\,\left[x\,y\right]\=\mathrm{Region}\left(x_L..x_R\,y_{B..}{y}_T\right)\right)$ ⇒ ${\int }_{x_L}^{x_R}{\int }_{y\=y_B\left(x\right)}^{y\=y_T\left(x\right)}\mathrm{\lambda } \mathrm{dy} \mathrm{dx}$

$\mathrm{int}\left(\mathrm{\λ}\,\left[y\,x\right]\=\mathrm{Region}\left(y_B..y_T\,x_L..x_R\right)\right)$ ⇒ ${\int }_{y_B}^{y_T}{\int }_{x\=x_L\left(y\right)}^{x\=x_R\left(y\right)}\mathrm{\lambda } \mathrm{dx} \mathrm{dy}$

Table 6.3.4   Syntax for the Maple commands that will implement a surface integral; $\mathrm{\lambda }\=\sqrt{1\+f_x^2\+f_y^2}$ 

Example 6.3.1

$F\=x y$; $R$ is the finite region bounded by the graph of $y\=x \left(1-x\right)$ and the $x$-axis.

See Example 6.2.1.

Example 6.3.2

$F\=2 x^2\+3 y^2$; $R$ is the finite region bounded by the graphs of $x\=y^2$ and $y\=3-2 x$.

See Example 6.2.2.

Example 6.3.3

$F\=2 x\+3 y\+1$; $R$ is the region bounded by the graphs of $f\left(x\right)\=\sin \left(x\right)$ and $g\left(x\right)\=\sin \left(2 x\right)$ on $0\le x\le \mathrm{\π}$. See Example 6.2.3.

Example 6.3.4

$F\=3 x^2\+2 y^2\+1$; $R$ is the region bounded by the graphs of $f\left(x\right)\=\arctan \left(x\+1\right)-1\/2$ and $g\left(x\right)\=\sin \left(x\right)$ on the interval $x\in \left[0\,\mathrm{\π}\/2\right]$. See Example 6.2.4.

Example 6.3.5

$F\=5-3 x^2-2 y^2$; $R$ is the interior of the triangle whose vertices are $\left(0\,0\right)\,\left(1\,0\right)\,\left(0\,1\right)$. See Example 6.2.5.

Example 6.3.6

$F\=\left(x-3\right) \left(y-6\right)$; $R$ is the interior of the triangle whose vertices are $\left(1\,3\right)\,\left(7\,4\right)\,\left(5\,9\right)$. See Example 6.2.6.

Example 6.3.7

$F\=2-{\left(y-1\/2\right)}^2$; $R$ is the region bounded by the graphs of 1, $\cos \left(x\right)$, and $y\=x$ on $0\le x\le 1$. See Example 6.2.7.

Example 6.3.8

$F\=\sqrt{2-x^2-4 y^2}$; $R$ is the interior of the ellipse $x^2\+4 y^2\=1$. See Example 6.2.8.

Example 6.3.9

Obtain the surface integral of $g\left(x\,y\right)\=x y$ over the part of the top half of the ellipsoid $3 x^2\+5 y^2\+7 z^2\=1$ that sits above the disk ${\left(x-1\/6\right)}^2\+{\left(y-1\/5\right)}^2\le 1\/5^2$.
See Example 6.2.9.

Example 6.3.10

Obtain the surface integral of $g\left(x\,y\right)\=x y$ over the part of the top half of the ellipsoid $3 x^2\+5 y^2\+7 z^2\=1$ that sits above the rectangle $0\le x\le 3\/10\,0\le y\le 1\/10$.

See Example 6.2.9.

Example 6.3.11

Obtain the surface integral of $g\left(x\,y\right)\={\left(x y\right)}^2$ over the part of the top half of the ellipsoid $3 x^2\+5 y^2\+7 z^2\=1$ that sits above the region bounded by the ellipse $3 x^2\+5 y^2\=1\/2$.

See Example 6.2.9.

Example 6.3.12

Obtain the surface integral of $g\left(x\,y\right)\=x y$ over the part of the top half of the ellipsoid $3 x^2\+5 y^2\+7 z^2\=1$ that sits above the first-quadrant region bounded by the ellipse

$3 x^2\+5 y^2\=1\/2$. See Example 6.2.9.

Example 6.3.13

Obtain the surface integral of $g\left(x\,y\right)\=x y$ on the surface $z_1\=7-x^2-y^2$defined over the plane region $R$ that is bounded by the curves $y\=x$, $y\=2-x^2$, and $x\=0$. See Example 6.2.10.

Example 6.3.14

Derive the expression for $\mathrm{d\σ}$ when the surface is given explicitly. See Table 6.3.2.

Example 6.3.15

Derive the expression for $\mathrm{d\σ}$ when the surface is given implicitly. See Table 6.3.2.

Example 6.3.16

Derive the expression for $\mathrm{d\σ}$ when the surface is given parametrically. See Table 6.3.2.