This is used to handle the problem of finding areas under a curve. For continuous functions, the Riemann sum on the right hand side represents the sum of the areas of n rectangles below the curve.

We all know as n increase, the widths of the rectangles approach zero, the errors in the approximations diminish, and the limit is the exact area under the curve.

So far so good. Now we will do now is that we will apply the exact same concepts but this time along two variables. Can it be done, well certainly. And to quicken our process in learning, I would suggest that the reader immediately be drawn to the parallels this idea when two dimensions are used. Here are hints to get your imagination going.

Instead of integration with respect to x, we now integrate with respect to an area. Instead of considering , we consider . Instead of finding areas, we are finding volumes. Now on to the formal definitions.

Just like how we started with a function and did the limiting process along one variable, that is graphically along the x-axis, in single variable calculus, we now perform our analysis with a function , defined in two variables, and we allow these two variables to vary. As you have guessed it, these two variables would form an area R on the x-y plane. Our limiting process, in three clear steps, is now done in this area.

For now, let us assume that the entire region R can be enclosed within some suitably large rectangle with sides parallel to the coordinate axes, meaning to say that R can’t be stretched to infinity. We say R is bounded. Lastly, let  be a function of two variables defined for all  in R.

Step 1:
Using lines parallel to the coordinate axes, divide the rectangle enclosing the region R into subrectangles, and exclude from consideration all those subrectange that contain any points outside of R. This leaves only rectangles that are subsets of R. Denote the areas of these rectangles

Step 2:
Choose an arbitrary point in each of these subrectangles, and denote them by

 *

As shown below, we can think of the product  as the volume of a rectangular parallelepiped with base area  and the height , so the sum

can be viewed as an approximate to the volume V of the entire solid with the base R stretch vertically up to . If this helps you visualize the concept, then that’s good. We’ll talk more about geometrical interpretation later.

Step 3:
There are two sources of error in the approximation. We want to remove these errors like how we removed the errors caused in using rectangles to approximate the area. First, the parallelepipeds have flat tops, whereas the surface  may be curved. Second, the rectangles that form the bases of the parallelepipeds do not completely cover the region R. To remove these errors, we create more subdivisions causing the lengths and widths of the base rectangles to approach zero plausibly implying that the errors of both types approach zero. The sum will therefore approach a limit L.

The sums are called Riemann sums, and the limit is denoted by

which is called the double integral of f(x,y) over R.

*Almost all notation is written  and not the usual . Up to now, I don’t know what the * is for. But for consistency sake, we shall keep it this way.

The most common way is to interpret this double integral is to envision the graph of  as a surface about the x-y plane. As mentioned during step 2, for each rectangle, the number  is the area of the rectangle parallelepiped. The sum of these parallelepipeds approximates the volume of the solid and by taking the limit, we get the exact volume.

It now becomes clear to many students the contrast of one variable and two variable calculus. In one variable, our function  represents a curve in a 2D plane and we integrate along an axis to get the area. Now, our function  represents a surface in a 3D space and we integrate with respect to an area to get a volume.

Our result is under the assumption that  is continuous and non-negative on the region R. In the case where  can be both positive and negative, the double integral over R represent the difference between volume of the solid above R and the that below R. We can this difference the net signed volume between  and R.

As usual, there comes certain properties with double integrals.

If  is nonnegative on the region R, then subdividing R into two regions  and  results in dividing the solid between R and  into two solids, the sum of which is the volume of the entire solid. This result is illustrated as

I would say that thinking of the double integral as a means of finding the volume between a surface and the area R suffices for our subsequent lessons, I wish to spur the more enthusiast students in seeing that this double integral can actually mean much more.

See, for each point on the curve specified by , we get a certain value of  given by . It is natural to think in terms of volumes because the graph of the surface illustrates just that; x and y represents the base and z represents the height. However, we can alternatively think the curve  as giving us a certain value for each combination of x and y. We are basically representing a quantity such as say mass, money, molecules, force or probability for each . Summing all these quantities, the double integral, gives us the sum of this quantity over the range specified by x and y, the area. This carries a much deeper meaning to the double integral, which I personally believe proves useful in its later applications.

Next lesson involves evaluating this double integral.