The general form of Leibniz’s Integral Rule with variable limits states:

Suppose satisfies the condition stated previously for the basic form of Leibniz’s Rule (LR-1, see “A Semi-Rigorous Derivation of Leibniz’s Rule“) . In addition, are defined and have continuous derivatives for Then for

(1) can be derived as a consequence of LR-1, the Multivariable Chain Rule,and the Fundamental Theorem of Calculus (FTC):

Let be continuous and have a continuous derivative in a domain of plane that includes the rectangle

I will derive LR-1 semi-rigorously as follows:

Let

Integrate (1-1) with respect to from a constant to a variable , we have

That is,

While and are functions of , is a constant.

Since

differentiate (1-2) with respect to gives

i.e.,

In the three-dimensional -space, the double integral of a continuous function with two independent variables, , may be interpreted as a volume between the surface and the -plane:

Fig. 1

We see from Fig. 1 that on one hand,

but on the other hand,

Since (2-1) and (2-2) amounts to the same thing, it must be true that

In other words, the order of integration can be interchanged.

“I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.” (Richard P. Feynman, “Surely You’re Joking, Mr. Feynman!”, Bantam Book, 1985)

“Feynman’s Trick” is a powerful technique for evaluating nontrivial definite integrals. It is based on Leibniz’s rule (LR-1) which states:

Let be a differentiable function in with continuous. Then

This is how it works in practice:

To evaluate definite integral

we introduce into integrand a parameter such that

when

and

when

Suppose

By Leibniz’s rule,

Integrate (4) with respect to :

where

Let

Let

And so,

Now, let’s play “Feynman’s Trick” on definite integral

It was known long ago that , the ratio of the circumference to the diameter of a circle, is a constant. Nearly all people of the ancient world used number for . As an approximation obtained through physical measurements with limited accuracy, it is sufficient for everyday needs.

An ancient Chinese text (周髀算经,100 BC) stated that for a circle with unit diameter, the ratio is .

In the Bible, we find the following description of a large vessel in the courtyard of King Solomon’s temple:

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high, It took a line of thirty cubits to measure around it. (1 Kings 7:23, New International Version)

This infers a value of .

It is fairly obvious that a regular polygon with many sides is approximately a circle. Its perimeter is approximately the circumference of the circle. The more sides the polygon has, the more accurate the approximation.

To find an accurate approximation for , we inscribe regular polygons in a circle of diameter . Let denotes the side’s length of regular polygon with and sides respectively,

Fig. 1

From Fig. 1, we have

It follows that

Substituting (4) into (1) yields

That is,

Further simplification gives

Starting with an inscribed square , we compute from (see Fig. 2). The perimeter of the polygon with sides is .

Fig. 2

Clearly,

.

Exercise-1 Explain, and then make the appropriate changes:

In “Mathematical Models in Biology”, Leah Edelstein-Keshet presents a model describing the number of circulating red blood cells (RBC’s). It assumes that the spleen filters out and destroys a fraction of the cells daily while the bone marrow produces a amount proportional to the number lost on the previous day:

Plotting (5) by ‘plot2d(4/3 + (-1)^(n+1)*2^(-n)/3, [n, 0, 10], WEB_PLOT)’ fails (see Fig. 1) since plot2d treats (5) as a continuous function whose domain includes number such as .

Fig. 1

Instead, a discrete plot is needed:

Fig. 2

From Fig. 2 we see that converges to a value between and . In fact,