Last time I defined a function, the zeta function, and the infinite sum. This raised a lot of questions, some of which I intend to answer in this column.

I will introduce another reference for the project:  Richard A. Silverman, Modern Calculus and Analytic Geometry, Macmillan Company, 1969 (republished in 2002 by Dover Publications). [1]  I looked up infinity and worked out this explanation: infinity is the name of the symbol ∞, and it represents a quantity that can be arbitrarily large. Another way of saying it is that no matter what numerical value you choose for ∞, you can always find a value that is larger.  Thus, ∞ is not a number, though it can be used in expressions as if it were.

We can apply this idea to our explanation of infinite sums as a sum whose number of terms never ends.  This raises the question:

41. If an infinite sum is one with a never-ending number of terms, how can you calculate it?

Using reference [2] we gain the definition: "Integration is the process of finding an integral." This is rather unsatisfactory. We now have the question,

42. What is an integral?

This seems pretty important so we will continue to consider this question.

According to [2] an integral is an area or the generalization of an area and is one of the fundamental concepts of calculus. The Riemann integral is the most commonly used type of integral. This raises the questions,

43. What is area?

44. What is the Riemann integral?

45. What other types of integrals are there?

According to [2] the area of some surface is the amount of material required to cover it completely.

According to [1] we can define an interval as the difference between two values of something.  For example, if we have and as two values of the increment of would be the difference of the latter value of subtracted by the previous value. We can write this,

If we have a function, and we assign a point, , on the increment , we can define the function at the point .  In a normal graph we can see that the vertical axis is while the horizontal is .  Let us say that the function we are studying is .  We then have.

If we think about it long enough it will occur to us that at any point is a length from a point on the axis to the curve.

We can also see that the interval describes a line segment along the axis. If we multiply this by we get a rectangular area.

This area has the value,

If we add up a number of these rectangles, we will get an approximation of the area between the curve and the axis.

The only problem with this method is that we get something that looks like a staircase.

There is a built-in error where the rectangles get close to the curve of the function. The accuracy of this method will increase if we make the width of each rectangle arbitrarily small. We can write this,

This means that the edge of the two increments is touching,

This is an important concept and requires a little more explanation at this point. Here the two points are touching, so there is no distance between them. But it does not mean that the two points occupy the same location. If we draw a little circle around each point,

we can say that the true location of each point is within the appropriate circle, called the neighborhood of the point.  If we say that these neighborhoods surround the values on the axis, we can say that there are corresponding neighborhoods about every point on the axis.  As the value of approaches some specific target (in this case 0), then will also approach some specific value called the limit of the function.  We write this,

It turns out that this is the definition of the Riemann integral. We can use standard notation. If we want to find the area under the curve of a function, , from the point to , we write it,

These facts raise a number of questions:

46. How do we use limits?

47. How do we use this idea to calculate integrals?

Richard A. Silverman, Modern Calculus and Analytic Geometry, Macmillan Company, 1969 (republished in 2002 by Dover Publications).

This is one of the nicest calculus books around. It has very clear descriptions, a nice set of problems to work, and has a nice blend of theory and applications. It may seem expensive for a Dover book (nearly $40), but this book has over a thousand pages and is split into 15 chapters that cover such topics as set theory and functions, real numbers, limits and continuity, derivatives, the applications of derivatives to different types of functions, integrals, plane analytic geometry, plane curves and plane vectors,  linear algebra, spatial analytic geometry, multivariable derivatives (called partial derivatives), multivariable integrals (called multiple integrals), applications of multivariable calculus, and infinite series. I do think that more emphasis should be placed on infinite series, and there is not much on differential equations. These problems so not keep me from recommending the book.

References

[1] Richard A. Silverman, Modern Calculus and Analytic Geometry, Macmillan Company, 1969 (republished in 2002 by Dover Publications).

[2] Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 1999.

Created by Mathematica  (November 21, 2004)