Double Integrals, Center of Mass Calculus IV Lab Karen E. Donnelly Saint Joseph's College LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQ=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

We can compute double integrals as iterated integrals in Maple just as we would by hand: LUkkaW50RzYiNiQtRiM2JComSSJ4R0YkIiIkSSJ5R0YkIiIjL0YrOyokRilGLCIiJS9GKTsiIiFGLA== With the MultiInt command in the Student[MultivariateCalculus] package we can cut down on the typing and number of parentheses required: LUkpTXVsdGlJbnRHNiI2JSomSSJ4R0YkIiIkSSJ5R0YkIiIjL0YpOyokRidGKiIiJS9GJzsiIiFGKg==

Consider the iterated integral of the function 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 over the region lying in the first quadrant bounded below by the parabola 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, and above by the horizontal line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4=. Maple has difficulties with the order of integral 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: LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtRiM2JComSSJ4R0YnIiIkLUkkc2luR0YnNiMqJEkieUdGJ0YtIiIjL0YyOyokRixGMyIiJS9GLDsiIiFGMw== However, reversing the order of integration to 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 helps. We can see why by computing the inner integral first: PkkmSW5uZXJHNiItSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JComSSJ4R0YkIiIkLUkkc2luR0YnNiMqJEkieUdGJEYtIiIjL0YsOyIiIS1JJXNxcnRHRic2I0Yy This has a computable antiderivative with respect to y: PkkzaW5kZWZpbml0ZWludGVncmFsRzYiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJJklubmVyR0YkSSJ5R0Yk Thus the final answer is now computatble by Maple: PkkkQU5TRzYiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJJklubmVyR0YkL0kieUdGJDsiIiEiIiU= If we wish to have a 10 digit approximation to this exact value, then LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJJEFOU0c2Ig== The combined command would just be: QyQtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1GJDYkKiZJInhHRigiIiQtSSRzaW5HRiU2IyokSSJ5R0YoRi4iIiMvRi07IiIhLUklc3FydEdGJTYjRjMvRjM7RjciIiUiIiI=LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= Sometimes reversing the order of integration will not allow one to solve a particular double integral analytically. When this happens, a numerical approximation method must be used. JSFH

Some regions in the plane that are not easily used as the region of integration of iterated integrals in rectangular coordinates. Switching to an integral in polar coordinates may make the problem much easier. Rules for polar integrals: The double polar integral should be expressed as follows: 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 1. Draw a sketch of the region R of integration. 2. Interpret this region in polar coordinates. 3. Shoot a ray from the pole (origin) out through the region. The entry point into the region R becomes the lower limit for the inner integral. The exit point from the region R becomes the upper limit for the inner integral. These limits MAY depend on \316\270, but not r. 4. The outer limits should be the minimum to maximum values of \316\270 that generate the entire region R. 5. Don't forget the r in the integrand ! (" 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 JSFH Example of Calculating Volume with Double Polar Integral As an example of using Maple to solve integration problems using polar coordinates: 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, 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, consider the following problem: Integrate the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieEYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRjZGOUZARkBGQC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmVvLUkmbXNxcnRHRiQ2Iy1GIzYpLUkjbW5HRiQ2JFEiNEYnRkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZkcC1GIzYkLUklbXN1cEdGJDYlRmVuLUZdcDYkUSIyRidGQC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQC1GPTYtUSIrRidGQEZCRkVGR0ZJRktGTUZPRmNwRmVwLUYjNiQtRmlwNiVGXm9GW3FGXnFGQEYrRkBGQEYrRkA= over the region R which lies in the first quadrant bounded between the two circles: 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 and 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. and the lines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInhGJ0Y0RjdGPkYrRj4=. First we plot the function with slices to help visualize the portion of the surface we are considering for the volume under the surface over the region. PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJC1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywoIiIlIiIiKiQ5JCIiIyEiIiokOSVGN0Y0RiRGJEYk QyQ+SSZTdXJmMUc2Ii1JJ3Bsb3QzZEdGJTYmLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YsOyIiISIiIy9GLUYvL0klYXhlc0dGJUkmYm94ZWRHRiUiIiI=SSIlRzYi QyQ+SSdTbGljZXNHNiItSSdwbG90M2RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2KDwkIiIiIiIjL0kmdGhldGFHRiU7IiIhLCRJI1BpR0YpI0YtIiIlL0kiekdGJTshIiIkIiNERjovSSdjb29yZHNHRiVJLGN5bGluZHJpY2FsR0YlL0kmc3R5bGVHRiVJLFBBVENITk9HUklER0YlL0klYXhlc0dGJUkmYm94ZWRHRiVGLQ==SSIlRzYi LUkoZGlzcGxheUc2IjYkPCRJJ1NsaWNlc0dGJEkmU3VyZjFHRiQvSSVheGVzR0YkSSZCT1hFREdGJA== To visualize the region R of integration, we plot it using Maple and rectangular coordinates: QyQ+SSNQMUc2Ii1JJXBsb3RHRiU2JC1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmIiIiRjAqJEkieEdGJSIiIyEiIi9GMjsqJC1GKjYjRjNGNEYwRjQ= QyQ+SSNQMkc2Ii1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JC1JJXNxcnRHRig2IywmIiIlIiIiKiRJInhHRiUiIiMhIiIvRjM7LUYtNiNGNEY0RjU= QyQ+SSNQM0c2Ii1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JEkieEdGJS9GLDsqJC1JJXNxcnRHRig2IyIiIyEiIkYwRjQ= LUkoZGlzcGxheUc2IjYlPCVJI1AxR0YkSSNQMkdGJEkjUDNHRiQvSSV2aWV3R0YkNyQ7IiIhIiIjRi0vSShzY2FsaW5nR0YkSSxjb25zdHJhaW5lZEdGJA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Or, in polar coordinates (easier) QyQ+SSthbmdsZWxpbmUxRzYiLUkqcG9sYXJwbG90R0YlNiU3JUkickdGJSIiIS9GKjsiIiIiIiMvSSZjb2xvckdGJUkkcmVkR0YlL0kqdGhpY2tuZXNzR0YlIiIkISIiJSFH QyQ+SSthbmdsZWxpbmUyRzYiLUkqcG9sYXJwbG90R0YlNiU3JUkickdGJSwkSSNQaUclKnByb3RlY3RlZEcjIiIiIiIlL0YqO0YvIiIjL0kmY29sb3JHRiVJJHJlZEdGJS9JKnRoaWNrbmVzc0dGJSIiJCEiIg== QyQ+SShjaXJjbGUxRzYiLUkqcG9sYXJwbG90R0YlNiU3JSIiIkkmdGhldGFHRiUvRis7IiIhLCRJI1BpRyUqcHJvdGVjdGVkRyNGKiIiJS9JJmNvbG9yR0YlSSRyZWRHRiUvSSp0aGlja25lc3NHRiUiIiQhIiI= QyQ+SShjaXJjbGUyRzYiLUkqcG9sYXJwbG90R0YlNiU3JSIiI0kmdGhldGFHRiUvRis7IiIhLCRJI1BpRyUqcHJvdGVjdGVkRyMiIiIiIiUvSSZjb2xvckdGJUkkcmVkR0YlL0kqdGhpY2tuZXNzR0YlIiIkISIi LUkoZGlzcGxheUc2IjYjPCZJKGNpcmNsZTFHRiRJKGNpcmNsZTJHRiRJK2FuZ2xlbGluZTFHRiRJK2FuZ2xlbGluZTJHRiQ= Setting up this double iterated integral in Cartesian coodindates would be difficult because you would have to break the region R into simpler subregions. It is very simple in polar coordinates. In polar coordinates the integral is the following: QyQtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1GJDYkKiYtSSVzcXJ0R0YlNiMsJiIiJSIiIiokSSJyR0YoIiIjISIiRjJGNEYyL0Y0O0YyRjUvSSZ0aGV0YUdGKDsiIiEsJEkjUGlHRiYjRjJGMUYyLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiJUdGJw== or, in stages it would be: QyQ+STBJbmRlZmluaXRlSW5uZXJHNiItSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JComLUklc3FydEdGKDYjLCYiIiUiIiIqJEkickdGJSIiIyEiIkYyRjRGMkY0RjI=QyQ+SSZJbm5lckc2Ii1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiYtSSVzcXJ0R0YoNiMsJiIiJSIiIiokSSJyR0YlIiIjISIiRjJGNEYyL0Y0O0YyRjVGMg==PkkmSW5uZXJHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiM= QyQ+STBJbmRlZmluaXRlT3V0ZXJHNiItSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JEkmSW5uZXJHRiVJJnRoZXRhR0YlIiIiPkkmT3V0ZXJHNiItSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JEkmSW5uZXJHRiQvSSZ0aGV0YUdGJDsiIiEsJEkjUGlHRigjIiIiIiIl The MultiInt command verifies our work. Note if we use the output = integral option -- the integral is shown unevaluated (this is called the inert form in Maple) -- to evaluate we would then use the value command. QyQtSSlNdWx0aUludEc2IjYmKiYtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJiIiJSIiIiokSSJyR0YlIiIjISIiRjBGMkYwL0YyO0YwRjMvSSZ0aGV0YUdGJTsiIiEsJEkjUGlHRisjRjBGLy9JJ291dHB1dEdGJUkpaW50ZWdyYWxHRiVGMA==QyQtSSZ2YWx1ZUdJKF9zeXNsaWJHNiI2I0kiJUdGJiIiIg==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiJUdGJw==

1. Evaluate the double integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJ5RidGL0YyLUkjbW5HRiQ2JFEiMkYnRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjk= over the region R in the first quadrant that is outside the circle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj5GPkYrRj4= and inside the cardiod 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. Plot of the region R for this exercise: Portion in first quadrant outside the circle and inside the cartiod. LUkqcG9sYXJwbG90RzYiNiU8JCIiIy1JIipHJSpwcm90ZWN0ZWRHNiRGJywmIiIiRi0tSSRjb3NHNiRGKkkoX3N5c2xpYkdGJDYjSSZ0aGV0YUdGJEYtL0YzOyIiISwkSSNQaUdGKkYnL0kqdGhpY2tuZXNzR0YkIiIk QyUtSSlNdWx0aUludEc2IjYmKiZJInJHRiUiIiQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmdGhldGFHRiUiIiMvRig7RjEtSSIqR0YtNiRGMSwmIiIiRjgtSSRjb3NHRixGL0Y4L0YwOywkSSNQaUdGLSMhIiJGMSwkRj4jRjhGMS9JJ291dHB1dEdGJUkpaW50ZWdyYWxHRiVGOC1JJnZhbHVlR0YuNiNJIiVHRiU= 2. Find the volume V of the solid above the region R = {(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEoJnRoZXRhO0YnL0YwRj1GOUY5)| LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4=, 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 } and under the surface 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. Plot of the region R for this exercise -- portion of disk in first quadrant colored blue: QyQ+SSNQMUc2Ii1JKnBvbGFycGxvdEdGJTYnPCQiIiIiIiQvSSZ0aGV0YUdGJTsiIiEsJEkjUGlHJSpwcm90ZWN0ZWRHI0YqIiIlL0koc2NhbGluZ0dGJUksY29uc3RyYWluZWRHRiUvSSZjb2xvckdGJUklQkxVRUdGJS9JKnRoaWNrbmVzc0dGJUYrRio=QyQ+SSNQMkc2Ii1JKnBvbGFycGxvdEdGJTYnPCQiIiIiIiQvSSZ0aGV0YUdGJTssJEkjUGlHJSpwcm90ZWN0ZWRHI0YqIiIlLCRGMCIiIy9JKHNjYWxpbmdHRiVJLGNvbnN0cmFpbmVkR0YlL0kmY29sb3JHRiVJJFJFREdGJS9JKnRoaWNrbmVzc0dGJUYrRio=LUkoZGlzcGxheUc2IjYjPCRJI1AxR0YkSSNQMkdGJA== QyctSSlNdWx0aUludEc2IjYmKiZJInJHRiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyokRigiIiNGKS9GKDtGKSIiJC9JJnRoZXRhR0YlOyIiISwkSSNQaUdGLSNGKSIiJS9JJ291dHB1dEdGJUkpaW50ZWdyYWxHRiVGKS1JJnZhbHVlR0YuNiNJIiVHRiVGKS1JJmV2YWxmR0YtRkI= JSFH

JSFH The Student[MultivariateCalculus] package has a built in command for computing the coordinates for the center of mass of a lamina with a given density function (the first argument to the command). Here we compute the center of mass of a lamina with density function 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 and the rectangular [1..4, -1..6] region representing the lamina (from Maple's help on CenterOfMass example): LUktQ2VudGVyT2ZNYXNzRzYiNiYsJiokSSJ4R0YkIiIjIiIiKiRJInlHRiRGKUYqL0YoO0YqIiIlL0YsOyEiIiIiJy9JJ291dHB1dEdGJEkpaW50ZWdyYWxHRiQ= LUktQ2VudGVyT2ZNYXNzRzYiNiUsJiokSSJ4R0YkIiIjIiIiKiRJInlHRiRGKUYqL0YoO0YqIiIlL0YsOyEiIiIiJw== The "output = plot command" gives the output in graphical form -- showing the region and the surface representing the density function, and the point representing the center of mass. LUktQ2VudGVyT2ZNYXNzRzYiNiYsJiokSSJ4R0YkIiIjIiIiKiRJInlHRiRGKUYqL0YoO0YqIiIlL0YsOyEiIiIiJy9JJ291dHB1dEdGJEklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJA== We can verify these calculations using the formulas for mass and the center of mass as a double integral: Calculate the mass m of the lamina: PkkibUc2Ii1JKU11bHRpSW50R0YkNiUsJiokSSJ4R0YkIiIjIiIiKiRJInlHRiRGK0YsL0YqO0YsIiIlL0YuOyEiIiIiJw== PkkjTXlHNiItSSlNdWx0aUludEdGJDYlKiZJInhHRiQiIiIsJiokRikiIiNGKiokSSJ5R0YkRi1GKkYqL0YpO0YqIiIlL0YvOyEiIiIiJw== PkkjTXhHNiItSSlNdWx0aUludEdGJDYlKiZJInlHRiQiIiIsJiokSSJ4R0YkIiIjRioqJEYpRi5GKkYqL0YtO0YqIiIlL0YpOyEiIiIiJw== QyQ+SSVDT014RzYiKiZJI015R0YlIiIiSSJtR0YlISIiRig=PkklQ29teUc2IiomSSNNeEdGJCIiIkkibUdGJCEiIg== The 2nd Moment (Moment of Inertia) about the x-axis -- This is a measure of the tendency of the lamina to resist rotation about the x-axis. (This is not a built-in function in Maple). It is computed 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, where \317\201 is the density function. For the example lamina above: PkkjSXhHNiItSSlNdWx0aUludEdGJDYlKiZJInlHRiQiIiMsJiokSSJ4R0YkRioiIiIqJEYpRipGLkYuL0YtO0YuIiIlL0YpOyEiIiIiJw== 2nd Moment (Moment of Inertia) about the y-axis -- This is a measure of the tendency of the lamina to resist rotation about the y-axis. It is computed as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ki1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvRjxRJ2l0YWxpY0YnLUYsNiVRImRGJ0ZYRmVuLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistRiM2Ki1GMjYnRjQtRiM2Ji1GLDYlUSJhRidGWEZlbi1GNTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y7RkFGREZGRkhGSkZMRk5GUEZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieUYnRlhGZW5GO0Y7RjstRiM2Ji1GLDYlUSJiRidGWEZlbkZpb0ZccEY7RmpuRl1vRistRiM2Jy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGWEZlbi1JI21uR0YkNiRGXG9GOy9GW29GX28tRjU2LVExJkludmlzaWJsZVRpbWVzO0YnRjtGQUZERkZGSEZKRkxGTkZQRlMtRiM2Ji1GLDYlUSYmcmhvO0YnL0ZZRkNGO0Zpby1GXXA2JC1GIzYmRl5xLUY1Ni1RIixGJ0Y7RkEvRkVGWkZGRkhGSkZMRk5GUC9GVFEsMC4zMzMzMzMzZW1GJ0ZhcEY7RjtGO0YrRjtGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0Zdcy8lKmxpbmVicmVha0dRJWF1dG9GJy1GNTYvUTAmRGlmZmVyZW50aWFsRDtGJ0Y4RjtGPkZBRkRGRkZIRkpGTEZORlBGU0ZecUY7RitGaHJGZnNGYXBGO0YrRjs=, where \317\201 is the density function. For the example lamina above: PkkjSXlHNiItSSlNdWx0aUludEdGJDYlKiZJInhHRiQiIiMsJiokRilGKiIiIiokSSJ5R0YkRipGLUYtL0YpO0YtIiIlL0YvOyEiIiIiJw== JSFH

Exercise 1. Calculate the center of mass for a lamina bounded by 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 and the y axis with density 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. (Use the CenterOfMass command to calculate and plot, then verify using MultiInt commands, as done above.) LUktQ2VudGVyT2ZNYXNzRzYiNiZJInhHRiQvRiY7IiIhLCYiIzsiIiIqJEkieUdGJCIiIyEiIi9GLjshIiUiIiUvSSdvdXRwdXRHRiRJJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ= LUktQ2VudGVyT2ZNYXNzRzYiNiZJInhHRiQvRiY7IiIhLCYiIzsiIiIqJEkieUdGJCIiIyEiIi9GLjshIiUiIiUvSSdvdXRwdXRHRiRJKWludGVncmFsR0Yk LUktQ2VudGVyT2ZNYXNzRzYiNiVJInhHRiQvRiY7IiIhLCYiIzsiIiIqJEkieUdGJCIiIyEiIi9GLjshIiUiIiU= Calculations using the formulas for mass and the center of mass as a double integral: Calculate the mass m of the lamina: PkkibUc2Ii1JKU11bHRpSW50R0YkNiUqJkkia0dGJCIiIkkieEdGJEYqL0YrOyIiISwmIiM7RioqJEkieUdGJCIiIyEiIi9GMjshIiUiIiU= PkkjTXlHNiItSSlNdWx0aUludEdGJDYlKiZJImtHRiQiIiJJInhHRiQiIiMvRis7IiIhLCYiIztGKiokSSJ5R0YkRiwhIiIvRjM7ISIlIiIl PkkjTXhHNiItSSlNdWx0aUludEdGJDYlKihJInlHRiQiIiJJImtHRiRGKkkieEdGJEYqL0YsOyIiISwmIiM7RioqJEYpIiIjISIiL0YpOyEiJSIiJQ== QyQ+SSVDT014RzYiKiZJI015R0YlIiIiSSJtR0YlISIiRig=PkklQ29teUc2IiomSSNNeEdGJCIiIkkibUdGJCEiIg== Exercise 2 Calculate the 2nd Moment (Moment of Intertia) about a) the y axis and then b) about the x axis for the lamina in Exercise 1. a) PkkjSXhHNiItSSlNdWx0aUludEdGJDYlKihJInlHRiQiIiNJImtHRiQiIiJJInhHRiRGLC9GLTsiIiEsJiIjO0YsKiRGKUYqISIiL0YpOyEiJSIiJQ== b) PkkjSXlHNiItSSlNdWx0aUludEdGJDYlLUkiKkclKnByb3RlY3RlZEc2JComSSJrR0YkIiIiSSJ4R0YkIiIjRi8vRi87IiIhLCYiIztGLiokSSJ5R0YkRjAhIiIvRjc7ISIlIiIl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

plot3d From the plots package polarplot From the Student[MultivariateCalculus] package MultiInt CenterOfMass