Triple Integrals Calc IV Lab Karen Donnelly Saint Joseph's College All rights reserved LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==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

Let Q be a bounded solid region in three space, and let f a function of three variables defined on Q. The triple integral integral of f over the solid region can defined in a manner similar to the definition of a double integral (see text). When f is the constant function 1, the triple integral gives us the volume of Q. Example: Let the region Q be the intersection of two right circular cylinders of radius one. The first is symmetric about the x-axis -- thus has equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNilGKy1GIzYlLUklbXN1cEdGJDYlLUYsNiVRInlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIyRicvRj9RJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGRS1JI21vR0YkNi1RIitGJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZTLyUpc3RyZXRjaHlHRlMvJSpzeW1tZXRyaWNHRlMvJShsYXJnZW9wR0ZTLyUubW92YWJsZWxpbWl0c0dGUy8lJ2FjY2VudEdGUy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlxvLUYjNiUtRjY2JS1GLDYlUSJ6RidGO0Y+RkFGR0ZKRkVGK0ZKRkUtRk42LVEiPUYnRkVGUUZURlZGWEZaRmZuRmhuL0Zbb1EsMC4yNzc3Nzc4ZW1GJy9GXm9Gam8tRkI2JFEiMUYnRkVGSkZFRitGSkZF. The second is symmetric about the z-axis -- thus has equation 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. We plot the cylinders: QyQ+SSNRMUc2Ii1JJ3Bsb3QzZEdGJTYoNyVJInhHRiUtSSRjb3NHRiU2I0kmdGhldGFHRiUtSSRzaW5HRiVGLS9GLjsiIiEsJEkjUGlHJSpwcm90ZWN0ZWRHIiIjL0YqOyEiI0Y3L0koc2NhbGluZ0dGJUksY29uc3RyYWluZWRHRiUvSSZzdHlsZUdGJUkscGF0Y2hub2dyaWRHRiUvSSVheGVzR0YlSSdub3JtYWxHRjYhIiI= QyQ+SSNRMkc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYoNyUtSSRjb3NHRig2I0kmdGhldGFHRiUtSSRzaW5HRihGL0kiekdGJS9GMDsiIiEsJEkjUGlHRikiIiMvRjM7ISIjRjkvSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JJnN0eWxlR0YlSSp3aXJlZnJhbWVHRiUvSSVheGVzR0YlSSdub3JtYWxHRikhIiI= LUkqZGlzcGxheTNkRzYiNiM8JEkjUTFHRiRJI1EyR0Yk Let us find the volume of the solid Q formed by the intersection of these two cylinders. Note that Q is made up of eight identical regions, one in each octant, so we can just compute the volume of the portion which lies in the first octant, plotted as follows: QyQ+SSNRM0c2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYoNyVJInhHRiUtSSRjb3NHRig2I0kmdGhldGFHRiUtSSRzaW5HRihGMC9GMTsiIiEsJEkjUGlHRikjIiIiIiIjL0YtO0Y2RjsvSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JJnN0eWxlR0YlSSxwYXRjaG5vZ3JpZEdGJS9JJWF4ZXNHRiVJJ25vcm1hbEdGKSEiIg== QyQ+SSNRNEc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYoNyUtSSRjb3NHRig2I0kmdGhldGFHRiUtSSRzaW5HRihGL0kiekdGJS9GMDsiIiEsJEkjUGlHRikjIiIiIiIjL0YzO0Y2RjsvSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JJnN0eWxlR0YlSSp3aXJlZnJhbWVHRiUvSSVheGVzR0YlSSdub3JtYWxHRikhIiI= LUkqZGlzcGxheTNkRzYiNiM8JEkjUTNHRiRJI1E0R0Yk To determine the appropriate limits for the triple iterated integral, there are six possible orders of integration: dz dx dy, dx dy dz, etc. If we assume the order is dz dx dy, and the region is "simple" with respect to this order (Fubini's Theorem 13.4), we can use an use the following rules to determine the limits of integration 1) Shoot a representative vertical line (parallel to the z axis) through the solid region Q. The entry point is the lower limit and the exit point is the upper limit for the innermost (dz) integral -- these limits may be function of both x and y. 2) Project the solid Q onto the x-y plane and then use the rules for double integrals to determine the limits for for the outer two integrals in terms of this projected region R. That is , if using dx dy, then: a) Shoot a horizontal line through the projected region R. The entry and exit point for this line determine the lower and upper regions for the second integral (middle) which may depend on y. b) For the outer integral limits (dy), choose the min to max values of y that include the entire projected region. Applying the rules to the above solid Q gives us The volume of this piece is 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. PkksUGllY2VWb2x1bWVHNiItSSRpbnRHRiQ2JC1GJjYkLUYmNiQiIiIvSSJ6R0YkOyIiIS1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywmRixGLCokSSJ5R0YkIiIjISIiL0kieEdGJEYvL0Y5O0YwRiw= If the Student MultiVariateCalculus package is loaded we can use the MultiInt form to save typing and parentheses confusion for the same computation: PkksUGllY2VWb2x1bWVHNiItSSlNdWx0aUludEdGJDYmIiIiL0kiekdGJDsiIiEtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJkYoRigqJEkieUdGJCIiIyEiIi9JInhHRiRGKy9GNTtGLEYo and thus the volume of the entire solid is eight times that of the piece just calculated. PkksVG90YWxWb2x1bWVHNiIsJEksUGllY2VWb2x1bWVHRiQiIik= Note: We could also have used a double integral to find the volume under the surface 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over the region R being the quarter of the unit circle in the first quadrant in the x-y plane. PkksUGllY2VWb2x1bWVHNiItSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JC1GJjYkLUklc3FydEdGJzYjLCYiIiJGMSokSSJ5R0YkIiIjISIiL0kieEdGJDsiIiFGLS9GMztGOUYx PkknVm9sdW1lRzYiLCRJLFBpZWNlVm9sdW1lR0YkIiIp

We can use triple integrals to find the mass, center of mass, and moments of inertia for solid objects. Using the solid Q as the same intersection of the two cylinders above, let T be the top half of the solid Q, and let the density of T at any point (x,y,z) be equal to the square of the distance from the point to the x-y plane, i.e., 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 . Compute the mass of T using the formula 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. PkkmVG1hc3NHNiIsJC1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLUYnNiQtRic2JCokSSJ6R0YkIiIjL0YxOyIiIS1JJXNxcnRHRig2IywmIiIiRjoqJEkieUdGJEYyISIiL0kieEdGJEY0L0Y8O0Y1RjoiIiU= Now to find the center of mass of T: Using symmetry, we can see that the x and y coordinates of the center of mass will both be 0. To determine the z coordinate, we first must find the first moment of T about plane x-y plane. PkkpTW9tZW50eHlHNiItSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JC1GJjYkLUYmNiQqJEkiekdGJCIiJC9GMDsiIiEtSSVzcXJ0R0YnNiMsJiIiIkY5KiRJInlHRiQiIiMhIiIvSSJ4R0YkOywkRjVGPUY1L0Y7O0Y9Rjk= The z coordinate of the center of mass is this moment divided by the mass of T: PkklemJhckc2IiomSSlNb21lbnR4eUdGJCIiIkkmVG1hc3NHRiQhIiI= Thus the center of mass of T is located at (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). We can also use the MultivariateCalculus package CenterOfMass command to do this calculation: LUktQ2VudGVyT2ZNYXNzRzYiNicqJEkiekdGJCIiIy9JInhHRiQ7LCQtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJiIiIkY0KiRJInlHRiRGKCEiIkY3Ri0vRjY7RjdGNC9GJzsiIiFGLS9JJ291dHB1dEdGJEkpaW50ZWdyYWxHRiQ= LUktQ2VudGVyT2ZNYXNzRzYiNiYqJEkiekdGJCIiIy9JInhHRiQ7LCQtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJiIiIkY0KiRJInlHRiRGKCEiIkY3Ri0vRjY7RjdGNC9GJzsiIiFGLQ==

Just as it is sometimes convenient to use polar coordinates when computing double integrals over appropriate regions, it is sometimes easier to use cylindrical coordinates to simplify the calculation of triple integrals. Recall that cylindrical coordinates and rectangular coordinates are just like polar coordinates, except we now have a z-coordinate which remains unchanged when converting between rectangular and cylindrical coordinates. (see section 10.7) 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, 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRRjEvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZURj4=. When evaluating triple integrals with cylindrical coordinates that we use 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. First example: As an example consider the solid Q formed below the plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4= and above the upper half of the right circular cone 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. Lets plot this cone using cylinderplot. Converting the equation of the cone to cylindrical form we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInJGJ0Y0RjcvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZXRj4= Since cylinderplot expects the first argument to be an expression for r as a function of z and theta: LUktY3lsaW5kZXJwbG90RzYiNiZJInpHRiQvSSZ0aGV0YUdGJDsiIiEsJEkjUGlHJSpwcm90ZWN0ZWRHIiIjL0YmO0YqIiIiL0klYXhlc0dGJEkmYm94ZWRHRiQ= The volume of Q using cylindrical coordinates 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. PkkoVm9sdW1lUUc2Ii1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLUYmNiQtRiY2JEkickdGJC9JInpHRiQ7Ri8iIiIvRi87IiIhRjMvSSZ0aGV0YUdGJDtGNiwkSSNQaUdGKCIiIw== LUkpTXVsdGlJbnRHNiI2JkkickdGJC9JInpHRiQ7RiYiIiIvRiY7IiIhRiovSSZ0aGV0YUdGJDtGLSwkSSNQaUclKnByb3RlY3RlZEciIiM= A second example: Convert the following triple integral in rectangular coordinates to one in cylindrical coordinates: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ky1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiM2Ji1GNTYtUSomdW1pbnVzMDtGJ0Y4RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GVS1JI21uR0YkNiRRIjJGJ0Y4LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y4RlcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GIzYrLUYyNidGNC1GIzYmRlEtSSZtc3FydEdGJDYjLUYjNigtRlg2JFEiNEYnRjgtRjU2LVEoJm1pbnVzO0YnRjhGO0Y+RkBGQkZERkZGSEZURlYtRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnRlcvRmluRl1vRmVuRjhGK0ZlbkY4RmVuRjhGZG9GaG5GW29GKy1GIzYqLUYyNidGNC1GWDYkUSIwRidGOC1GIzYqRmlvRlxwRl9wRlxwLUYjNiUtRmJwNiUtRiw2JVEieUYnRmdwRmpwRldGXHFGZW5GOEYrRmVuRjhGaG5GW28tSShtZmVuY2VkR0YkNiQtRiM2KUYrRl9wLUY1Ni1RIitGJ0Y4RjtGPkZARkJGREZGRkhGVEZWRmZxRitGZW5GOEY4RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGanIvJSpsaW5lYnJlYWtHUSVhdXRvRictRjU2LVEwJkRpZmZlcmVudGlhbEQ7RidGOEY7Rj5GQEZCRkRGRkZIRkpGTS1GLDYlUSJ6RidGZ3BGanBGZW5GOEYrRmVyRmNzRmpxRmVuRjhGK0ZlckZjc0ZkcEZlbkY4RitGZW5GOA==. We can see that Maple can handle the rectangular coordinates: LUkpTXVsdGlJbnRHNiI2JiwmKiRJInhHRiQiIiMiIiIqJEkieUdGJEYpRiovSSJ6R0YkOyIiISwoIiIlRipGJyEiIkYrRjMvRiw7LCQtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJkYyRipGJ0YzRjNGNy9GKDshIiNGKQ== However if we had to do it by hand, it would be much easier in cylindrical coordinates. This is a triple integral of the function 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over the solid Q bounded above by the paraboloid LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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and bounded below by the plane z = 0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=. Plot this solid Q of integration with cylinderplot. The first argument to cylinderplot represents the radius r given in terms of the coordinates theta and z. Converting the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=to cylindrical form gives 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= or solving for r in terms of z: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=. Thus we use: LUktY3lsaW5kZXJwbG90RzYiNiYtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJiIiJSIiIkkiekdGJCEiIi9JJnRoZXRhR0YkOyIiISwkSSNQaUdGKSIiIy9GLztGNEYtL0klYXhlc0dGJEkmYm94ZWRHRiQ= The integral in cylindrical coordinates becomesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=. LUkpTXVsdGlJbnRHNiI2JyokSSJyR0YkIiIkL0kiekdGJDsiIiEsJiIiJSIiIiokRiciIiMhIiIvRic7RixGMS9JJnRoZXRhR0YkO0YsLCRJI1BpRyUqcHJvdGVjdGVkR0YxL0knb3V0cHV0R0YkSSlpbnRlZ3JhbEdGJA== LUkmdmFsdWVHSShfc3lzbGliRzYiNiNJIiVHRiU=

1. a) Using a triple integral with rectangular coordinates (and the order dz dy dx), find the volume of the three dimensional solid below the elliptic paraboloid given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiotSSNtbkdGJDYkUSMxMkYnRj4tRjs2LVEoJm1pbnVzO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0Zobi1GIzYoLUZXNiRRIjJGJ0Y+LUY7Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGY28tRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGNEY3RlxvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPkYrRmBwRj5GWi1GIzYoRlxvRl9vLUYjNiUtRmhvNiUtRiw2JVEieUYnRjRGN0Zcb0ZdcEZgcEY+RitGYHBGPkYrRmBwRj5GK0ZgcEY+RitGYHBGPg==and above z = 0 Below is a plot to help you visualize this solid and the limits of integration: LUkvaW1wbGljaXRwbG90M2RHNiI2Ky9JInpHRiQsKCIjNyIiIiokSSJ4R0YkIiIjISIjKiRJInlHRiRGLUYuL0YsOyEiJSIiJS9GMEYyL0YnOyIiISIjOS9JJWdyaWRHRiQ3JSIjP0Y9Rj0vSSZjb2xvckdGJEkobWFnZW50YUdGJC9JJnN0eWxlR0YkSSxwYXRjaG5vZ3JpZEdGJC9JJWF4ZXNHRiRJJ25vcm1hbEclKnByb3RlY3RlZEcvSS10cmFuc3BhcmVuY3lHRiQkIiImISIi Replace WHAT's with appropriate values 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 b) Calculate this triple integral again using cylindrical coordinates. 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 c) Calculate the volume again, now using double integrals and rectangular coordinates (calculating volume as double integral over region under surface). 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 d) Calculate the volume again, now using double integrals and polar coordinates (calculating volume as double integral over region under surface). 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 2. Using triple integrals in rectangular coordinates, and the order dz dx dy, find the mass and center of mass of the solid bounded by the planes y = -3, y = 3, z = 0 and by the surface 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assuming that the density function is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUYjNictRiw2JVEmJnJobztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGOS8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlAtSShtZmVuY2VkR0YkNiQtRiM2KS1GLDYlUSJ4RicvRjdRJXRydWVGJy9GOlEnaXRhbGljRictRj02LVEiLEYnRjlGQC9GQ0ZmbkZERkZGSEZKRkxGTi9GUlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGZW5GZ25GaW4tRiw2JVEiekYnRmVuRmduLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y5RjlGZW9GOS1GPTYtUSI9RidGOUZARkJGREZGRkhGSkZML0ZPUSwwLjI3Nzc3NzhlbUYnL0ZSRlxwLUYjNiktSSVtc3VwR0YkNiVGWC1JI21uR0YkNiRRIjJGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGOUZARkJGREZGRkhGSkZMRk5GUS1GYXA2JUZfb0ZjcEZncEZqcC1GYXA2JUZiby1GZHA2JFEiM0YnRjlGZ3BGZW9GOUYrRmVvRjlGK0Zlb0Y5. This is the portion of the solid cylinder with generating curve 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 which is "cut off" at z = 0 and also at y = -3 and y = 3. Thus it is the solid under the surface plotted below: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Same CenterOfMass command as above but leave out "output =integral" to see value: JSFH