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

Reference: 

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.

Reference:

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

Review of some basic notions of double integrals

 I am back in this blog after 3 months of interruption to continue educating about Calculus. In this post I am going to review the following notions: volume and double integral, properties of double integrals, Fubini's theorem, iterated integral, applications of double integrals.

Volume and double integral

 Concept:

The approximation of the volume of a solid bounded above by a function of two variables over a rectangular region is used to define the notion of double integral. We use the same approach to define the simple integral by calculating the area of a rectangle under a curve. 

We start by dividing the solid in smaller and smaller solids. At each iteration, we calculate volume of the solid. As the number of solids becomes infinitely large, the volume tends to a fixed number. This number represents the volume of the solid. It is impossible to enumerate all the volumes obtained especially when the number of smaller solids becomes bigger and bigger. The notion of limit intervenes here since as as the number of smaller solids becomes bigger and bigger  the volume of the entire stays fixed. This limit is the limit of the double Rieman sum. This limit is also defined as the double integral of the function f(x,y) over the region R. We write:

Definition

The double integral of a function function f(x,y) over a rectangular region R in the xy plane is defined as the limit of the double Rieman sum as written above.

Reference:

Double integrals of a function f(x,y) over a rectangular region R

Properties of double integrals

Concept:

Properties of double integrals are useful to simplify the computations and find values on their bounds.

Reference:

Properties of double integrals

Fubini's theorem

Concept:

Fubni's theorem is used to write and evaluate a double integral as an iterated integral

If f(x.y) is a function of two variables that is continuous over a rectangular region R:

Then the double integral of f over the region equals an iterated integral

Iterated Integral

Concept:

The iterated integral allows to calculate the double integral by focusing on one integral at a time. Here are some properties of the iterated integral.

Reference:

Double integrals of a function f(x,y)  over a rectangular region R :

Applications of double integral

Concept:

Double integrals can be used  to calculate the area of a region, the volume under a surface and the average value of a function of two variables over rectangular region.

Reference:

Calculating the area of a rectangular region and the volume of a solid using double integrals