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?&#x2018;&#x201C;(đ?&#x2018;Ľ, đ?&#x2018;Ś) over đ??ˇ is defined as the Integral of the Extended Function đ?&#x2018;&#x201C; â&#x2C6;&#x2014; đ?&#x2018;Ľ, đ?&#x2018;Ś over the Extended rectangle đ?&#x2018;&#x2026;

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How to calculate Double Integrals defined over General Regions?

By the Cavalieriâ&#x20AC;&#x2122;s Method (The Slide Method)

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Taking Normal Slides to "đ?&#x2018;Ľ" or "đ?&#x2018;Ś" Axes

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And calculating the Volume as the one-dimensional integral of Slides Areas

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Generalized Fubiniâ&#x20AC;&#x2122;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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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 â&#x20AC;&#x201C; 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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4.3_4.4_Integral_Doble_Dominios_Generales_Polares

Calculus II Topic 4: Multiple Integrals Calculus II Santiago de Vicente 1 Santiago de Vicente 2 Calculus II Santiago de Vicente 3 Calculus I...