Exam 14: Section 3: Multiple Integration

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Given Given   use polar coordinates to set up and evaluate the double integral   . use polar coordinates to set up and evaluate the double integral Given   use polar coordinates to set up and evaluate the double integral   . .

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E

Use a double integral to find the area of the region inside the circle Use a double integral to find the area of the region inside the circle   and outside the cardioid   . Round your answer to two decimal places. and outside the cardioid Use a double integral to find the area of the region inside the circle   and outside the cardioid   . Round your answer to two decimal places. . Round your answer to two decimal places.

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C

Use a double integral to find the area enclosed by the graph of Use a double integral to find the area enclosed by the graph of   .  . Use a double integral to find the area enclosed by the graph of   .

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C

Use a double integral in polar coordinates to find the volume of the solid in the first octant bounded by the graphs of the equations given below. Use a double integral in polar coordinates to find the volume of the solid in the first octant bounded by the graphs of the equations given below.

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Evaluate the following iterated integral by converting to polar coordinates. Evaluate the following iterated integral by converting to polar coordinates.

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Use a double integral to find the area enclosed by the graph of Use a double integral to find the area enclosed by the graph of   .  . Use a double integral to find the area enclosed by the graph of   .

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Evaluate the double integral below. Evaluate the double integral below.

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Evaluate the following iterated integral by converting to polar coordinates. Evaluate the following iterated integral by converting to polar coordinates.

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Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations   if one-tenth of the volume of the solid is removed. Round your answer to four decimal places. if one-tenth of the volume of the solid is removed. Round your answer to four decimal places.

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Suppose the population density of a city is approximated by the model Suppose the population density of a city is approximated by the model   where x and y are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. Round your answer to the nearest integer. where x and y are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. Round your answer to the nearest integer.

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Evaluate the double integral below. Evaluate the double integral below.

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Evaluate the iterated integral Evaluate the iterated integral   by converting to polar coordinates. Round your answer to four decimal places. by converting to polar coordinates. Round your answer to four decimal places.

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Combine the sum of the two iterated integrals into a single integral by converting to polar coordinates. Evaluate the resulting iterated integral. Combine the sum of the two iterated integrals into a single integral by converting to polar coordinates. Evaluate the resulting iterated integral.

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Use polar coordinates to describe the region as shown in the figure below: Use polar coordinates to describe the region as shown in the figure below:

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Use a double integral in polar coordinates to find the volume of the solid inside the hemisphere Use a double integral in polar coordinates to find the volume of the solid inside the hemisphere   but outside the cylinder   . but outside the cylinder Use a double integral in polar coordinates to find the volume of the solid inside the hemisphere   but outside the cylinder   . .

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Identify the region of integration for the following integral. Identify the region of integration for the following integral.

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Evaluate the double integral below. Evaluate the double integral below.

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Use a double integral to find the area of the shaded region as shown in the figure below. Use a double integral to find the area of the shaded region as shown in the figure below.

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Find a such that the volume inside the hemisphere Find a such that the volume inside the hemisphere   and outside the cylinder   is one-half the volume of the hemisphere. Round your answer to four decimal places. and outside the cylinder Find a such that the volume inside the hemisphere   and outside the cylinder   is one-half the volume of the hemisphere. Round your answer to four decimal places. is one-half the volume of the hemisphere. Round your answer to four decimal places.

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Evaluate the iterated integral Evaluate the iterated integral   by converting to polar coordinates. by converting to polar coordinates.

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