Chapter 6: Applications of Double Integration
Section 6.3: Surface Area
Calculate the surface area of the surface defined by the function F=x−3 y−6 whose domain is the plane region R, the interior of the triangle whose vertices are 1,3,7,4,5,9. See Example 6.2.6.
Using Figure 6.1.6(a) from Example 6.1.6, obtain the equations of the edges of the triangle R.
The surface is defined by Fx,y=x−3 y−6, so the surface-area element is
The surface area is then given by the iterated integral
∫15∫YABYACλⅆy ⅆx+∫57∫YABYBCλ dy dx = 39.06346474
where λ=1+y−62+x−32. The inner integrals can be evaluated in closed form, but then the outer integrals must be evaluated numerically.
Maple Solution - Interactive
The most direct calculation of the surface area over a triangular domain is implemented with the task template contained in Table 6.3.6(a).
Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a Triangle
Surface Integral on a Surface Defined over a Triangle
Table 6.3.6(a) Task-template implementation of the SurfaceInt command
A solution from first principles requires the equations of the edges of the triangle forming the region R. These equations were found in Example 6.1.6, and are again given in Table 6.3.6(b) where the Context Panel option Assign Name is applied.
Table 6.1.6(b) Equations of the edges of the triangle forming region R
The calculation of λ=1+Fx2+Fy2 appears in Table 6.3.5(c).
Context Panel: Assign Name
Expression palette: Square-root template
Calculus palette: Partial-derivative operator
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻lambda
1+∂∂ x F2+∂∂ y F2 = 1+y−62+x−32→assign to a nameλ
Table 6.3.5(c) Calculation of λ=1+Fx2+Fy2
The iterated integral by means of which the surface area over R can be found is given in Table 6.2.5(d).
Iterate in the order dy dx via the template in the Calculus palette
Calculus palette: Iterated double-integral template
Context Panel: 2-D Math≻Convert To≻Inert Form
Press the Enter key.
Context Panel: Approximate≻10 (digits)
∫15∫YABYACλ ⅆy ⅆx+∫57∫YABYBCλ ⅆy ⅆx
→at 10 digits
Table 6.2.5(d) Iterated double-integral for finding the surface area of F over R
Once again, the inner integrals are evaluated in closed form, but the outer integrals must be evaluated numerically.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the integrand F.
Obtain the equations of the edges in region R
Use the Line command from the Student Precalculus package.
YAB = x6+176
YBC = 432−5⁢x2
YAC = 3⁢x2+32
Use the diff command to obtain the requisite partial derivatives.
Top-level, using the Int and evalfcommands
Use the MultiInt command from the Student MultivariateCalculus package
Use the MultiInt command with a pre-defined domain option
MultiIntλ,x,y=Triangle1.,3,7.,4,5,9 = 39.06346476
Use the SurfaceInt command in the Student VectorCalculus package
<< Previous Example Section 6.3
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)