<Text-field style="Heading 1" layout="Heading 1">Motivating the Definition of the Double Integral</Text-field>

Double Integrals, Center of Mass

Calculus IV Lab

Karen E. Donnelly

Saint Joseph's College

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ==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

<Text-field style="Heading 1" layout="Heading 1">Motivating the Definition of the Double Integral</Text-field>

The Approximate Integration Tutor helps motivate the definition of the double integral (see page 991-992)

QyQtJiZJKFN0dWRlbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0k1TXVsdGl2YXJpYXRlQ2FsY3VsdXNHRio2I0k0QXBwcm94aW1hdGVJbnRUdXRvckdGKkYqIiIi

<Text-field style="Heading 1" layout="Heading 1">Computing Double Integrals with Maple</Text-field>

We can compute double integrals as iterated integrals in Maple just as we would by hand:

With the MultiInt command in the Student[MultivariateCalculus] package we can cut down on the typing and number of parentheses required:

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

Adding the option "output = integral" displays as an integral which can be evaluated with value(%)

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

Adding the option "output = steps" shows the step by step operations to carry out the integration

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

JSFH

<Text-field style="Heading 1" layout="Heading 1">An Example of When Reversing Order of Integration Helps</Text-field>

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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:

However, reversing the order of integration to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ki1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOC1GUDYkUSI0RidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIyRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrLUYjNiotRjI2J0Y0Rk8tSSZtc3FydEdGJDYjLUYsNiVRInlGJy8lJ2l0YWxpY0dRJXRydWVGJy9GOVEnaXRhbGljRidGVkZZRistRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGYG9GY28tRlA2JFEiM0YnRjgvRldGZW4tRjU2LVExJkludmlzaWJsZVRpbWVzO0YnRjhGO0Y+RkBGQkZERkZGSEZKRk0tRmhvNiUtRiM2Ji1GLDYlUSRzaW5GJy9GYW9GPUY4LUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjhGO0Y+RkBGQkZERkZGSEZKRk0tSShtZmVuY2VkR0YkNiQtRiM2JkYrLUYjNiQtRmhvNiVGXW9GXXBGYHBGOEYrRjhGOEY4LUZQNiRRIjJGJ0Y4RmBwRjhGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0Zgci8lKmxpbmVicmVha0dRJWF1dG9GJy1GNTYtUTAmRGlmZmVyZW50aWFsRDtGJ0Y4RjtGPkZARkJGREZGRkhGSkZNRmpvRjhGK0ZbckZpckZdb0Y4RitGOA== helps. We can see why by computing the inner integral first:

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

This has a computable antiderivative with respect to y:

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

Thus the final answer is now computatble by Maple:

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

If we wish to have a 10 digit approximation to this exact value, then

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJEFOU0YnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmhuRkBGK0ZobkZARitGaG5GQA==

The combined command would just be:

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

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.

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

<Text-field style="Heading 1" layout="Heading 1"> Double Integrals in Polar Coordinates</Text-field>

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:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVEmJmludDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRi1GIzYkLUYvNi1RIj9GJ0YyRjVGOEY6RjxGPkZARkIvRkVRLDAuMTExMTExMWVtRicvRkhGT0YyRkkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1JI21pR0YkNiNRIUYnRlctRi82LVExJkludmlzaWJsZVRpbWVzO0YnRjJGNUY4RjpGPEY+RkBGQkZERkctRlg2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYuLUZYNiVRInJGJ0Zbb0Zeby1GLzYtUScmc2RvdDtGJ0YyRjVGOEY6RjxGPkZARkJGREZHLUZYNiVRKmNvcyYjOTUyO0YnRltvRl5vLUYvNi1RIixGJ0YyRjUvRjlGXW9GOkY8Rj5GQEZCRkQvRkhRLDAuMzMzMzMzM2VtRictRi82LVEifkYnRjJGNUY4RjpGPEY+RkBGQkZERkdGZW9GaG8tRlg2JVEqc2luJiM5NTI7RidGW29GXm8vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUrZXhlY3V0YWJsZUdGXW8vJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjJGMkZob0Zlb0Zoby1GLzYtUTAmRGlmZmVyZW50aWFsRDtGJ0YyRjVGOEY6RjxGPkZARkJGREZHRmVvLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmpxLyUqbGluZWJyZWFrR1ElYXV0b0YnRmJxLUZYNiVRKCZ0aGV0YTtGJy9GXG9GN0YyRmpwRl1xRl9xRjI=

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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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

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

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.

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEnU2xpY2VzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYnLUYsNiVRJ3Bsb3QzZEYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGaW4tSShtZmVuY2VkR0YkNiQtRiM2MC1GXG82Ji1GIzYnLUkjbW5HRiQ2JFEiMUYnRkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0Zobi9GVVEsMC4zMzMzMzMzZW1GJy1GZW82JFEiMkYnRkAvJStleGVjdXRhYmxlR0ZERkBGQC8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGaG8tRiM2KC1GLDYlUScmIzk1MjtGJy9GN0ZERkAtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtRiM2Jy1GZW82JFEiMEYnRkAtRj02LVEjLi5GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRidGam4tSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEnJiM5NjA7RidGXnFGQEZhcEZALUYjNiUtRmVvNiRRIjRGJ0ZARmFwRkAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl5zLyUpYmV2ZWxsZWRHRkRGYXBGQEYrRmFwRkBGaG8tRiM2KC1GLDYlUSJ6RidGNkY5Rl9xLUYjNihGKy1GIzYmLUY9Ni1RKiZ1bWludXMwO0YnRkBGQkZFRkdGSUZLRk1GT0ZqcS9GVUZbckZkb0ZhcEZARmdxLUZlbzYkUSQyLjVGJ0ZARmFwRkBGK0ZhcEZARmhvLUYjNictRiw2JVEnY29vcmRzRidGNkY5Rl9xLUYsNiVRLGN5bGluZHJpY2FsRidGNkY5RmFwRkBGaG8tRiM2Jy1GLDYlUSZzdHlsZUYnRjZGOUZfcS1GLDYlUSxQQVRDSE5PR1JJREYnRjZGOUZhcEZARmhvLUYjNictRiw2JVElYXhlc0YnRjZGOUZfcS1GLDYlUSZib3hlZEYnRjZGOUZhcEZARitGYXBGQEZARmFwRkBGK0ZhcEZARitGYXBGQC1GPTYtUSI7RidGQEZCRltwRkdGSUZLRk1GT0ZobkZURmFwRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSIlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

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

To visualize the region R of integration, we plot it using Maple and rectangular coordinates:

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

JSFH

Or, in polar coordinates (easier)

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSthbmdsZWxpbmUyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GIzYnLUYsNiVRKnBvbGFycGxvdEYnRjRGNy1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZ24tSShtZmVuY2VkR0YkNiQtRiM2Ki1Gam42Ji1GIzYqLUYsNiVRInJGJ0Y0RjctRjs2LVEiLEYnRj5GQC9GREY2RkVGR0ZJRktGTUZmbi9GU1EsMC4zMzMzMzMzZW1GJy1JJm1mcmFjR0YkNigtRiM2JS1GLDYlUScmIzk2MDtGJy9GNUZCRj4vJStleGVjdXRhYmxlR0ZCRj4tRiM2JS1JI21uR0YkNiRRIjRGJ0Y+RmRwRj4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmFxLyUpYmV2ZWxsZWRHRkJGZW8tRiM2KEZiby1GOzYtUSI9RidGPkZARkNGRUZHRklGS0ZNRk9GUi1GIzYnLUZpcDYkUSIxRidGPi1GOzYtUSMuLkYnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJ0Zobi1GaXA2JFEiMkYnRj5GZHBGPkYrRmRwRj5GK0ZkcEY+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGZW8tRiM2Jy1GLDYlUSZjb2xvckYnRjRGN0ZocS1GLDYlUSRyZWRGJ0Y0RjdGZHBGPkZlby1GIzYnLUYsNiVRKnRoaWNrbmVzc0YnRjRGN0ZocS1GaXA2JFEiM0YnRj5GZHBGPkYrRmRwRj5GPkZkcEY+RitGZHBGPi1GOzYtUSI6RidGPkZARkNGRUZHRklGS0ZNRk9GUkZkcEY+

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

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:

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

or, in stages it would be:

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

The MultiInt command verifies our work.

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

<Text-field style="Heading 1" layout="Heading 1"> Exercises for Double Polar Integrals</Text-field>

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiYtSSNtbkdGJDYkUSIyRidGPi1GOzYtUTEmSW52aXNpYmxlVGltZXM7RidGPkZARkNGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnL0ZTRmhuLUkobWZlbmNlZEdGJDYkLUYjNictRlc2JFEiMUYnRj4tRjs2LVEiK0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0Zmby1GIzYmLUYsNiVRJGNvc0YnL0Y1RkJGPi1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk1GZ25GaW4tRltvNiQtRiM2JC1GLDYlUSgmdGhldGE7RidGXXBGPkY+Rj5GPkYrRj5GPkY+RitGPkYrRj4=.

Plot of the region R for this exercise: Portion in first quadrant outside the circle and inside the cartiod.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEqcG9sYXJwbG90RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2MS1GVzYmLUYjNigtSSNtbkdGJDYkUSIyRidGQC1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRiM2JkZpbi1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPRlFGVC1GVzYkLUYjNictRmpuNiRRIjFGJ0ZALUY9Ni1RIitGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGY3AtRiM2Jy1GLDYlUSRjb3NGJy9GN0ZERkBGPC1GVzYkLUYjNiUtRiw2JVEnJiM5NTI7RidGanBGQC8lK2V4ZWN1dGFibGVHRkRGQEZARmJxRkBGK0ZARkBGQEYrRmJxRkBGQC8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGXW9GX3EtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yNzc3Nzc4ZW1GJy9GVUZeci1Gam42JFEiMEYnRkAtRj02LVEjLi5GJ0ZARkJGRUZHRklGS0ZNRk9GYnBGVEZpbkZlby1GLDYlUScmIzk2MDtGJ0ZqcEZARl1vLUYsNiVRKnRoaWNrbmVzc0YnRjZGOUZqcS1Gam42JFEiM0YnRkBGYnFGQEZARmJxRkBGK0ZicUZARitGYnFGQA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

2. Find the volume V of the solid above the region

R = {(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEoJnRoZXRhO0YnL0YwRj1GOUY5)| LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GPkYrRj4=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSUmbGU7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRKCZ0aGV0YTtGJy8lJ2l0YWxpY0dGPkY1RjUtRjk2LVEiLEYnRjVGPC9GQFEldHJ1ZUYnRkFGQ0ZFRkdGSS9GTFEmMC4wZW1GJy9GT1EsMC4zMzMzMzMzZW1GJy1GIzYmRlBGOC1JJm1mcmFjR0YkNigtRiM2JC1GLDYlUSUmcGk7RidGU0Y1RjUtRiM2JC1GMjYkUSI0RidGNUY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZccC8lKWJldmVsbGVkR0Y+RjVGK0Y1 } and under the surface 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.

Plot of the region R for this exercise -- portion of disk in first quadrant colored blue:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Calculating Center of Mass, Second Moments in Maple</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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):

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEtQ2VudGVyT2ZNYXNzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNjMtSSVtc3VwR0YkNiUtRiw2JVEieEYnRi9GMi1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiK0YnRkQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk8vJSlzdHJldGNoeUdGTy8lKnN5bW1ldHJpY0dGTy8lKGxhcmdlb3BHRk8vJS5tb3ZhYmxlbGltaXRzR0ZPLyUnYWNjZW50R0ZPLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGaG4tRjs2JS1GLDYlUSJ5RidGL0YyRkBGRi1GSjYtUSIsRidGREZNL0ZRRjFGUkZURlZGWEZaL0ZnblEmMC4wZW1GJy9Gam5RLDAuMzMzMzMzM2VtRidGPS1GSjYtUSI9RidGREZNRlBGUkZURlZGWEZaL0ZnblEsMC4yNzc3Nzc4ZW1GJy9Gam5GXHAtRkE2JFEiMUYnRkQtRko2LVEjLi5GJ0ZERk1GUEZSRlRGVkZYRlpGZm4vRmpuRmVvLUZBNiRRIjRGJ0ZERmBvRl1vRmhvLUZKNi1RKiZ1bWludXMwO0YnRkRGTUZQRlJGVEZWRlhGWkZmbkZpbkZecEZhcC1GQTYkUSI2RidGREZERkQtRiw2I1EhRicvJStleGVjdXRhYmxlR0ZPRkQ=

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.

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

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:

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

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:

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

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 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, where \317\201 is the density function. For the example lamina above:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Exercises for Center of Mass, Moments</Text-field>

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.)

Calculations using the formulas for mass and the center of mass as a double integral:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Calculate the mass m of the lamina:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMUY0

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Major Commands Used</Text-field>

plot3d

From the plots package

polarplot

From the Student[MultivariateCalculus] package

MultiInt

CenterOfMass