Chapter 7: Triple Integration
Section 7.5: Integration in Spherical Coordinates
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Example 7.6.2
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Use spherical coordinates to integrate the function over , the region above the cone but below the unit sphere that is centered at .
(This region was graphed in Example 7.5.4.)
(See Example 7.4.11 where this integral is evaluated in cylindrical coordinates.)
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Solution
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Mathematical Solution
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Iteration in the order leads to
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The astute reader will note that the value of the integral, namely , is the volume of the region . This region consists of a hemisphere of radius 1 (having volume ) and a cone (having volume ).
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use plots in
module()
local p1,p2,p3;
p1:=plot3d(2*cos(phi), theta=0..2*Pi,phi=0.. (1/4)*Pi,coords=spherical);
p2:=plot3d([rho, theta,Pi/4],rho=0.. sqrt(2),theta=0..2*Pi, coords = spherical); p3:=display(p1,p2, scaling=constrained, axes=frame,labels=[x,y,z],tickmarks=[3,3,3],orientation=[-30,80,0],lightmodel=none);
print(p3);
end module:
end use:
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Figure 7.6.2(a) The region
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The upper limit of the innermost integral is obtained by using to rewrite as and solving for .
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The upper limit of the middle integral is obtained by noting that for the cone , .
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Maple Solution - Interactive
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Table 7.6.2(a) provides a solution by a visualization task template.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Spherical
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Evaluate and Graph
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, where
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Table 7.6.2(a) Solution by visualization task template
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Table 7.6.2(b) provides a solution by a task template that embodies the MultiInt command from the Student MultivariateCalculus package. It iterates in just the order , although the command itself will iterate in any of the six possible orders.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Spherical
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Iterated Triple Integral in Spherical Coordinates
( = colatitude, measured down from -axis)
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Integrand:
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Region:
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Inert Integral:
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Value:
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Stepwise Evaluation:
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Table 7.6.2(b) Solution by task template that embodies the MultiInt command
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Access the MultiInt command via the Context Panel
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Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Fill in the fields of the two dialogs shown below.
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Context Panel: Evaluate Integral
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Table 7.6.2(c) contains a solution from first principles, with the iterated integral being set via the iterated triple-integral template in the Calculus palette.
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Calculus palette: Iterated triple-integral template
(In the integrand, be sure to include the Jacobian for spherical coordinates.)
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Context Panel: Evaluate and Display Inline
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Table 7.6.2(c) Solution from first principles
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Maple Solution - Coded
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Initialize
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Install the Student MultivariateCalculus package.
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Apply the MultiInt command from the Student MultivariateCalculus package
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Solution via the top-level Int and int commands
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