In cylindrical coordinates, the upper surface of is given by
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Context Panel: Assign Name
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while the lower surface is . The lateral surface is a cylinder whose cross section is the circle .
Table 7.4.10(a) provides a solution by a visualization task template. After selecting the order of iteration, the integrand and the fields for the limits of integration are given. The resulting value of the integral and a graph of the region of integration are generated by pressing the appropriate buttons.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical
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Evaluate and Graph
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Volume Element
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, where
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Table 7.4.10(a) Task template for iterating a triple integral in cylindrical coordinates
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Constrained scaling is applied to the graph via its Context Panel.
Table 7.4.10(b) provides a solution from first principles. Note that the Jacobian, , must be inserted into the integrand.
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Calculus palette:
Iterated triple-integral template
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Context Panel: Evaluate and Display Inline
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Table 7.4.10(b) Solution from first principles
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Table 7.4.10(c) provides a solution by a task template that uses the MultiInt command from the Student MultivariateCalculus package, and iterates in the order . The command takes a coordinate option, and hence, inserts the appropriate Jacobian automatically.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical
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Iterated Triple Integral in Cylindrical Coordinates
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Integrand:
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Region:
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Inert Integral:
(Note automatic insertion of Jacobian.)
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Value:
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Stepwise Evaluation:
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Table 7.4.10(c) Solution by task template that implements 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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