Chapter 6: Applications of Double Integration
Section 6.3: Surface Area
Example 6.3.13
Obtain the surface integral of on the surface defined over the plane region that is bounded by the curves , , and . See Example 6.2.10.
Solution
Mathematical Solution
The surface is defined by , so the surface-area element is
Iterating in the order results in the integral
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Maple Solution - Interactive
Initialize
Control-drag the equation .
Context Panel: Assign Name
The simplest approach is to employ the task template in Table 6.3.13(a).
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a 2-D Region
Surface Integral on a Surface Defined over a General 2-D Region
Integrand
Surface
Table 6.3.13(a) Solution by task template
A solution from first principles is given in Table 6.3.13(b).
Obtain
Calculus palette: Partial-derivative template
Write an appropriate iterated integral and evaluate
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
=
Table 6.3.13(b) Solution from first principles
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the bounding surface .
Use to diff command to obtain the partial derivatives with respect to and .
Form the integral via the MultiInt command with a pre-defined domain option
Evaluate the exact answer with the evalf command
Use the SurfaceInt command from the Student VectorCalculus package
A solution from first principles that uses the top-level Int command is given below.
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