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MI.3 Double Integrals over General Regions

For single integrals, the region over which we integrate is always an interval. But for double integrals, we want to be able to integrate a function not just over rectangles but also over regions D of more general shape. We suppose that D is a bounded region, which means that D can be enclosed in a rectangular region R. Then we define a new function F with domain R by:

Picture of D and D within R. See page 1015

If the double integral of F exists over R, the we define the double integral of over D by

The definition above makes sense because R is a rectangle and so $\int \int_{R}$ F has been previously defined in section 16.1. The procedure that we have used is reasonable because the values of are 0 when lies outside D. and so they contribute nothing to the integral. This means that it doesn't matter what rectangle R we use, as long as it contains D. This is show below:

Picture of region R, with D highlighted and with F above it. If possible, highlight the volume of the graph of F.

Type I Double Integrals

A plane region D is said to be of type I if it lies between the graphs of two continuous functions of x, that is,

Where $g_{1}$ and $g_{2}$ are continuous on Some examples of type I regions are show below:

Pictures of type one regions

In order to evaluate these integrals, we simply choose the bounds described by D. Therefore to set up an double integral of a type I general region:

Now we simply integrate.

Type II Double Integrals

A region of type II is described as follows:

Some examples of type II regions are shown below:

pictures of type 2 regions

Set up a type II double integral as follows:

Properties of Double Integrals

From the previous sections we already know the following properties:

If $f(x,y)\geq g(x,y).$

A new property is given below. This property is similar to the property of single integrals given by the equation

If $D=D_{1}\cup D_{2},$ where $D_{1}$ and $D_{2}$ don't overlap except perhaps on their boundaries:

picture of what is described above

Then:

The above property can be used to evaluate double integrals over regions D that are neither type I or type II but can be expressed as a union of regions of type I and type II.

The next property of integrals says that if we integrate the constant functions over a region D, we get the area of D:

Finally we can combine some of the previous properties to give the following property:

If $m\leq f(x,y)\leq M$ for all in in D, then

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