Chapter 7: Triple Integration
Section 7.1: The Triple Integral
In Maple, implement the definition of the triple integral for fx,y,z=1+2 x+3 y+5 z on the region R defined by the inequalities 1≤x,y,z≤2.
Define fx,y,z as the integrand
Context Panel: Assign Function
fx,y,z=1+2 x+3 y+5 z→assign as functionf
Form a Riemann sum
Expression palette: Summation template
Context Panel: Assign Name
q=∑i=1u∑j=1v∑k=1wf1+iu,1+jv,1+kw⋅1u v w→assign
Obtain an iterated limit
Apply the relevant form of the limit command.
limitq,u=∞,v=∞,w=∞ = 16
In three dimensions, Maple's limit command computes iterated limits, not a true multidimensional limit. This is in distinction to the two-dimensional case where, for certain functions, Maple can obtain a true bivariate limit. Note also that there is no convenient "syntax-free" way to implement the multidimensional limit.
The expression for the Riemann sum in three dimensions can be cumbersome. Its simplified form is displayed below.
simplifyq = 12⁢32⁢u⁢v⁢w+5⁢u⁢v+3⁢u⁢w+2⁢v⁢ww⁢u⁢v
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