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