Triple Integrals in Spherical Coordinates Calc IV lab Karen Donnelly Saint Joseph's College

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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

Spherical Coordinates: Below we graph the the sphere with radius one given by the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1JI21uR0YkNiRRIjNGJ0Y5LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y5RitGV0Y5RitGV0Y5, using the command plot3d with coords=spherical option. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSIlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

Here we graph the top half of a right circular cone cut off so that it has height 3. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSIlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9 Below we display the sphere and cone together: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

When calculating a volume of a solid region, the volume in spherical coordinates can be calculated as 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 . Visualize this order of integration as three sweeping motions, the first generates a line segment, then second generates a planar region, and the third sweeps out the solid. Consider the intersection of the cone and sphere plotted above. We can use the iterated integral formula to calculate its volume as 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 Using the MultiInt command from the Student[MultivariateCalculus] package: 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 Or alternatively using the Maple integrate command directly: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIlZGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZobkZARitGaG5GQEYrRmhuRkA= If we want to calculate the mass of a solid region, we integrate the density function over the region, of course converting the density to spherical coordinates. For the solid above, with a density constant density k, we calculate : 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 To calculate the center of mass of this solid region with constant density, we use the symmetry of the solid and the uniform density to conclude that the x and y coordinates of the center of mass are 0. Thus we only need to calculate the z coordinate of the center of mass. We first calculate the moment about the xy-plane -- where we use the conversion 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, so: 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 Then we calculate the z coordinate as : 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Exercise 1: The following plots a torus described in spherical coordinates by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEmJnJobztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPUYnRjkvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZQLUYjNigtSSNtbkdGJDYkUSI0RidGOS1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGOUZARkJGREZGRkhGSkZML0ZPUSYwLjBlbUYnL0ZSRmduLUYjNictRiw2JVEkc2luRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjlGQEZCRkRGRkZIRkpGTEZmbkZobi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJiZwaGk7RidGNkY5LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y5RjlGaW9GOUYrRmlvRjlGK0Zpb0Y5RitGaW9GOUYrRmlvRjk=. 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 Calculate its volume by setting up the appropriate triple integral: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Exercise 2: The following graphs the wrinkled sphere described in Section Project of Larson, page 1041 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 Calculate its volume by setting up the appropriate triple integral: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Exercise 3: The following graphs the bumpy sphere described in Section Project of Larson, page 1041 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 Calculate its volume by setting up the appropriate triple integral: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Bonus Exercises: Vary the parameters in the bumpy and wrinkled spheres or torus above to create different surfaces -- plot and calculate their volumes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

From the Student[MultivariateCalculus ]package: MultiInt