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).
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Surface Integration≻Surface Defined over a Triangle
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Surface Integral on a Surface Defined over a Triangle
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Integrand
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Surface
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Triangle
Vertices
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Table 6.3.6(a) Task-template implementation of the SurfaceInt command
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A solution from first principles requires the equations of the edges of the triangle forming the region . 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.
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Table 6.1.6(b) Equations of the edges of the triangle forming region
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The calculation of appears in Table 6.3.5(c).
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Context Panel: Assign Name
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Expression palette: Square-root template
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Calculus palette: Partial-derivative operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻lambda
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Table 6.3.5(c) Calculation of
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The iterated integral by means of which the surface area over can be found is given in Table 6.2.5(d).
Iterate in the order via the template in the Calculus palette
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Calculus palette: Iterated double-integral template
Context Panel: 2-D Math≻Convert To≻Inert Form
Press the Enter key.
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Context Panel: Approximate≻10 (digits)
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Table 6.2.5(d) Iterated double-integral for finding the surface area of over
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Once again, the inner integrals are evaluated in closed form, but the outer integrals must be evaluated numerically.