Chapter 6: Applications of Double Integration
Section 6.3: Surface Area
By way of review, Table 6.3.1 lists forms for the arc-length element ds along curves given either explicitly or parametrically.
By integrating the arc-length element ds, the length of a curve is established.
Similarly, Table 6.3.2 lists the surface-area element dσ for surfaces defined explicitly, implicitly, or parametrically.
The surface area S is then given by the double
Arc-Length Element (ds)
Table 6.3.1 Arc-length element ds
integral S= ∫∫R1 dσ, where the integral is taken over R, the region over which the surface is defined.
For surfaces given either explicitly or implicitly, dA is either dy dx or dx dy, depending on the order of iteration in the double integral S.
For surfaces given parametrically with parameters u and v, dA^is either du dv or dv du, depending on the order of iteration in the double integral of dσ over the support region R′ in the uv-plane.
Of course, for the explicitly given surface, the notation fx2 means the square of the partial derivative taken with respect to x, etc.
For the implicitly given surface, the notation ∇h stands for the (Euclidean) norm of the gradient of h, that is, for
while the notation hz stands for the absolute value of the partial with respect to z.
J1=∂y,z∂u,v = |yuyvzuzv|
J2=∂z,x∂u,v = |zuzvxuxv|
J3=∂x,y∂u,v = xuxvyuyv
Table 6.3.2 Surface-area element dσ
For the parametrically given surface, the expressions for Jk,k=1,2,3, are Jacobians, the determinants of the corresponding Jacobian matrices.
It is also possible to define "surface integrals" where the integrand is some function g defined on the surface. For example, g could represent a surface charge-density, and the surface integral of g over some surface would be the total charge on that surface.
As with arc length, the integrals arising in surface-area and surface-integral calculations typically cannot be evaluated in closed form. In some cases, the inner integral can be so evaluated, but the outer integral then has to be evaluated numerically.
Maple Tools for Surface Integration
Because all Maple's integration commands insert the differential elements, any discussion that references dσ is actually referencing dA as well. But Maple's integration commands insert the appropriate form of dA, so it is convenient to write dσ=λ dA (or dσ=λ dA^), where, in the third column of Table 6.3.2, λ points to that part of dσ not containing differentials.
Table 6.3.3 lists Maple's built-in tools that implement some form of surface integration.
Int and/or int commands at top level
User must obtain and insert the appropriate form of dσ′.
Surface area when integrand is 1; surface integral otherwise.
SurfaceArea command in the Student MultivariateCalculus package
Calculates and inserts dσ′.
Computes surface area only for explicitly given surfaces.
Admits polar coordinates.
Surface-area task template
Calculus - Multivariate≻Integration≻Surface Area
Implements the SurfaceArea command and iterates in the order dy dx
SurfaceInt command in the Student VectorCalculus package
Calculates surface area if integrand is set to 1, else calculates a surface integral for more general scalar functions defined on the surface.
Recognizes pre-defined regions known to the modified int command in the Student VectorCalculus package.
Table 6.3.3 Maple tools for surface integration
The SurfaceArea command in the Student MultivariateCalculus package can return the value of the surface area, the unevaluated integral giving that area; or when R, the planar region over which the surface is defined is a rectangle, a graph showing the surface and R.
Unfortunately, the syntax for the SurfaceArea command is not consistent with that of the top-level int command or with the MultiInt command. For both the top-level int command and the MultiInt command, if ranges are given in the order y=…,x=…, then the resulting iteration order is dy dx. However, for the SurfaceArea command, the opposite is true: if ranges are given in the order y=…,x=…, then the resulting iteration order is dx dy.
The SurfaceInt command in the Student VectorCalculus package can return the value of a surface integral, or the unevaluated iterated integral. The command recognizes the Box and Sphere options, and the more general Surface option. The Surface option takes the parametric form of a surface, given as a vector. The region R over which the surface is defined can be any one of the pre-determined two-dimensional regions recognized by the modified int command in the Student VectorCalculus package. In particular, these regions are the Circle, Ellipse, Rectangle, Triangle, Sector (of a circle or ellipse), and the more general Region. In addition, the command takes a coordinate option that applies to the vector giving the surface in parametric form.
Table 6.3.4 clarifies the syntax for using the commands with which a surface integral can be implemented in Maple. To make the expressions visually compact, the size of some fonts has been reduced, and the notation λ=1+fx2+fy2 introduced.
Intλ,y=…,x=… ⇒ ∫xLxR∫y=yBxy=yTxλ dy dx
Intλ,x=…,y=… ⇒ ∫yByT∫x=xLyx=xRyλ dx dy
MultiIntλ,y=…,x=… ⇒ ∫xLxR∫y=yBxy=yTx λ dy dx
MultiIntλ,x=…,y=… ⇒ ∫yByT∫x=xLyx=xRy λ dx dy
SurfaceAreaf,x=…,y=… ⇒ ∫xLxR∫y=yBxy=yTxλ dy dx
SurfaceAreaf,y=…,x=… ⇒ ∫yByT∫x=xLyx=xRyλ dx dy
SurfaceIntgx,y,x,y,z=Surfacex,y,fx,y,x=…,y=… ⇒ ∫xLxR∫y=yBxy=yTxgx,y λ dy dx
SurfaceIntgx,y,y,x,z=Surfacex,y,fx,y,y=…,x=… ⇒ ∫yByT∫x=xLyx=xRygx,y λ dx dy
(as modified in Vector Calculus)
intλ,x,y=RegionxL..xR,yB..yT ⇒ ∫xLxR∫y=yBxy=yTxλ dy dx
intλ,y,x=RegionyB..yT,xL..xR ⇒ ∫yByT∫x=xLyx=xRyλ dx dy
Table 6.3.4 Syntax for the Maple commands that will implement a surface integral; λ=1+fx2+fy2
In Examples 6.3.(1-8), calculate the surface area of the surface defined by the function F over the region R. (Each of these examples corresponds to an example in Section 6.2.)
F=x y; R is the finite region bounded by the graph of y=x 1−x and the x-axis.
See Example 6.2.1.
F=2 x2+3 y2; R is the finite region bounded by the graphs of x=y2 and y=3−2 x.
See Example 6.2.2.
F=2 x+3 y+1; R is the region bounded by the graphs of fx=sinx and gx=sin2 x on 0≤x≤π. See Example 6.2.3.
F=3 x2+2 y2+1; R is the region bounded by the graphs of fx=arctanx+1−1/2 and gx=sinx on the interval x∈0,π/2. See Example 6.2.4.
F=5−3 x2−2 y2; R is the interior of the triangle whose vertices are 0,0,1,0,0,1. See Example 6.2.5.
F=x−3 y−6; R is the interior of the triangle whose vertices are 1,3,7,4,5,9. See Example 6.2.6.
F=2−y−1/22; R is the region bounded by the graphs of 1, cosx, and y=x on 0≤x≤1. See Example 6.2.7.
F=2−x2−4 y2; R is the interior of the ellipse x2+4 y2=1. See Example 6.2.8.
Obtain the surface integral of gx,y=x y over the part of the top half of the ellipsoid 3 x2+5 y2+7 z2=1 that sits above the disk x−1/62+y−1/52≤1/52.
See Example 6.2.9.
Obtain the surface integral of gx,y=x y over the part of the top half of the ellipsoid 3 x2+5 y2+7 z2=1 that sits above the rectangle 0≤x≤3/10,0≤y≤1/10.
See Example 6.2.9.
Obtain the surface integral of gx,y=x y2 over the part of the top half of the ellipsoid 3 x2+5 y2+7 z2=1 that sits above the region bounded by the ellipse 3 x2+5 y2=1/2.
Obtain the surface integral of gx,y=x y over the part of the top half of the ellipsoid 3 x2+5 y2+7 z2=1 that sits above the first-quadrant region bounded by the ellipse
3 x2+5 y2=1/2. See Example 6.2.9.
Obtain the surface integral of gx,y=x y on the surface z1=7−x2−y2defined over the plane region R that is bounded by the curves y=x, y=2−x2, and x=0. See Example 6.2.10.
Derive the expression for dσ when the surface is given explicitly. See Table 6.3.2.
Derive the expression for dσ when the surface is given implicitly. See Table 6.3.2.
Derive the expression for dσ when the surface is given parametrically. See Table 6.3.2.
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