Triple Integrals in Spherical Coordinates Calc IV lab LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYmUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIjtGJ0Y3L0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==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LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlIVEkRiotJS5CT1VORFNfSEVJR0hURzYjJCIlZ0lGKi0lKUNISUxEUkVORzYi Spherical Coordinates: Below we graph the the sphere with radius one given by the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1JI21uR0YkNiRRIjNGJ0Y5RjlGK0Y5RitGOQ==, using the command plot3d with coords=spherical option. 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

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

When calculating a volume of a solid region, the formula for volume in spherical coordinates is . See figure 14.68 page 1038 to 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. The solid in Example 4, page 1039, is the intersection of the cone and sphere plotted above. We can use the iterated integral formula to calculate its volume as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1GIzYqLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVEmJmludDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1JI21uR0YkNiRRIjBGJ0Y8LUYjNiYtRlQ2JFEiMkYnRjwtRjk2LVExJkludmlzaWJsZVRpbWVzO0YnRjxGP0ZCRkRGRkZIRkpGTEZORlEtRiw2JVElJnBpO0YnLyUnaXRhbGljR0ZBRjxGPC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIyRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrLUYjNiotRjY2J0Y4RlMtSSZtZnJhY0dGJDYoLUYjNiRGaW5GPC1GIzYkLUZUNiRRIjRGJ0Y8RjwvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmdwLyUpYmV2ZWxsZWRHRkFGXm9GYW9GKy1GIzYqLUY2NidGOEZTLUZUNiRRIjNGJ0Y8Rl5vRmFvRistRiM2Jy1JJW1zdXBHRiQ2JS1GLDYlUSYmcmhvO0YnRlxvRjxGWS9GX29GY29GZm4tRiM2Ji1GLDYlUSRzaW5GJ0Zcb0Y8LUY5Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjxGP0ZCRkRGRkZIRkpGTEZORlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSkmdmFycGhpO0YnRlxvRjxGPEY8RjxGK0Y8RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGYXMvJSpsaW5lYnJlYWtHUSVhdXRvRictRjk2LVEwJkRpZmZlcmVudGlhbEQ7RidGPEY/RkJGREZGRkhGSkZMRk5GUUZocUY8RitGXHNGanNGaXJGPEYrRlxzRmpzLUYsNiVRKCZ0aGV0YTtGJ0Zcb0Y8RjxGK0Y8RitGPEYrRjw= 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 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. Thus as in Example 5, we calculate : 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 To calculate the center of mass of this solid region in the same Example 5, 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 , so : 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 Then we calculate the z coordinate as : 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

Exercise 1: The following plots the torus of Exercise 31: 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 Calculate its volume by setting up the appropriate triple integral: Exercise 2: The following graphs the wrinkled sphere described in Section Project, page 1041 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 Calculate its volume by setting up the appropriate triple integral: Exercise 3: The following graphs the bumpy sphere described in Section Project, page 973 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 Calculate its volume by setting up the appropriate triple integral: