Review of some basic notions about double integrals (continued)

In this post I am going to review the notions of double integrals over general regions.

General bounded region. Definition

A general bounded region D on the plane is a region that can be enclosed in a rectangular region

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General regions of integration

Calculation of an iterated integral over a general bounded region

To calculate an iterated integral over a general bounded region, we sketch the region and express it a type I or type II region or a union of several type I and type II regions that overlap only on their boundaries.

Key Equations

Reference:

Double integrals over non-rectangular regions

Volume, area and average volume of a function of two variables over general non rectangular regions

The volume, area and average value of a function of two variables over general non rectangular regions can be found the same way as for functions of two variable over rectangular regions.

Reference:

Using double integrals to calculate the volume of a solid over a general region

Improper double integral

An improper double integral is the double integral of a function of two variable over an unbounded region. We use Fubini's theorem to evaluate some types of improper integrals.

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Improper double integrals

Polar coordinates

Double integrals in polar coordinates can be used over a rectangular polar region or a general polar region.. We use an iterated integral similar to those in double integrals over  rectangular region in plane coordinates. To  convert from plane coordinates to polar cordinates use 

To convert from polar coordinates to rectangular coordinates use:

The volume of a  solid in polar coordinates bounded above by a surface z = f(r, Θ) over a region in the rectangular plane is found by double integrals in polar coordinates.

Reference:

Double integral over polar rectangular regions