Chapter 6: Applications of Double Integration
Section 6.3: Surface Area
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.
The surface is defined by Fx,y=7−21 x2−35⁢y2/7 , the top half of the ellipsoid 3 x2+5 y2+7 z2=1, so the surface-area element is
Iterating in the order dy dx results in the integral
∫0310∫0110x⁢y⁢17⁢12⁢x2+10⁢y2−73⁢x2+5⁢y2−1ⅆyⅆx ≐ 0.0002353725944
As a function of y, the integrand is sufficiently complicated that, although Maple can find an antiderivative for it, this antiderivative fills several pages. Hence, the iterated integral is evaluated numerically.
Maple Solution - Interactive
Solve 3 x2+5 y2+7 z2=1 for z=zx,y
Context Panel: Assign to a Name≻q
3 x2+5 y2+7 z2=1→assign to a nameq
Type the name q and press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻z
Context Panel: Assign to a Name≻Z
→solutions for z
→assign to a name
The simplest approach is to employ the task template in Table 6.3.10(a).
Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a Rectangle
Surface Integral on a Surface Defined over a Rectangle
Table 6.3.10(a) Task template for surface integration over a rectangle
A solution from first principles is given in Table 6.3.10(b).
Calculus palette: Partial-derivative template
Context Panel: Assign Name
λ=1+∂∂ x Z12+∂∂ y Z12→assign
Write an appropriate iterated integral and evaluate numerically
Calculus palette: Iterated double-integral template
Context Panel: 2-D Math≻Convert To≻Inert Form
Context Panel: Approximate≻10 (digits)
∫03/10∫01/10x y λ ⅆy ⅆx
→at 10 digits
Table 6.3.10(b) Solution from first principles
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the bounding surfaces zx,y=Z± by invoking the solve and eval commands.
q≔x−1/62+y−1/52=1/25:Z≔solve3 x2+5 y2+7 z2=1,z:
Use the diff command to obtain the partial derivatives with respect to x and y.
Apply the simplify command.
Form the integral via the MultiInt command with a pre-defined domain option
Evaluate the integral with the evalf command
S≔MultiIntx y λ,x,y=Rectangle0..3/10,0..1/10,output=integral
evalfS = 0.0002353725944
Use the SurfaceInt command from the Student VectorCalculus package
evalfQ = 0.0002353725944
A solution from first principles that uses the top-level Int command is given below.
Intx y λ,y=0..1/10,x=0..3/10=evalfIntx y λ,y=0..1/10,x=0..3/10
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