Lectures 25 and 26 Double Integrals over General Regions in Cartesian and Polar Coordinates Calculus II Topic 4: Multiple Integrals Calculus II
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First Approach: Internal (or external) Partition of a Bounded Region into Rectangular Cells
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Integral: Usual limit of Riemann Sums
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Second Approach: Defining a Extended by Zero Function over a Extended Rectangular Domain
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The Integral of đ?‘“(đ?‘Ľ, đ?‘Ś) over đ??ˇ is defined as the Integral of the Extended Function đ?‘“ ∗ đ?‘Ľ, đ?‘Ś over the Extended rectangle đ?‘…
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How to calculate Double Integrals defined over General Regions?
By the Cavalieri’s Method (The Slide Method)
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Taking Normal Slides to "đ?‘Ľ" or "đ?‘Ś" Axes
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And calculating the Volume as the one-dimensional integral of Slides Areas
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Generalized Fubini’s Theorem
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Summary:
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Simple Example
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Solving the Simple Example
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You can choose the Integration Order
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Working Example 1
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Working Example 2
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Remember (Last Week):
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Finding Integration Limits
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Calculating Areas by Double Integration
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You must choose the more interesting (efficient) Limits
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More Examples
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The Easiest Way to calculate the Volume
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Another Example of Volume Calculation with careful choice of Integration Limits
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Volume between two Surfaces
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General Properties of Double Integral
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Additivity Property
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It is common to find domains that are easily described in terms of Polar Coordinates that using Cartesian Coordinates
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In such cases it is useful to calculate the Double Integral in Polar Coordinates
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Riemman Double Integral in Polar Coordinates
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Area Element in Polar Coordinates
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For a General Domain of the Form:
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We have the Theorem of Change of Variables (Cartesian – Polar) for Double Integrals
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Remember:
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Example
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Finding the Limits of Integration in Polar Coordinates
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Finding the Limits of Integration in Polar Coordinates. EXAMPLE
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Finding the Limits of Integration in Polar Coordinates. EXAMPLE
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Area of a Semicircle
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Area of a Lemmniscate of Bernoulli
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Area between two Circles
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Common Area to two Circles
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Area between Lemniscate and Circle
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Homework: Area of the Figure (In both Cartesian and Polar Coordinates)
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Volumes in Polar Coordinates
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Homework: Volume in both Cartesian and Polar Coordinates
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Homework
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