(^238) Integrals
When an enlightened scientist must calculate the area ISI of S, he writes
(4.48) ISI = lim If(,) Ax = f a'f(x) dx
and, at least in simple cases, evaluates the integral with the aid of Theorem
4.38. The procedure by which (4.48) is obtained must now be explained.
The first step is to sketch an appropriate figure which will look more or
less like Figure 4.471 or Figure 4.472. The next step is to make a parti.
tion P of the interval a < x S b into subintervals, but we do not bother
to draw more than one of the subintervals. Without bothering with
subscripts and stars, we let Ax denote the length of the interval and let x
be a point of the interval. We then draw the rectangle whose width is
Ax and whose height is f(x). The first step in building the formula
(4.48) is to write f(x) Ax, because this is the area of the rectangle (or
rectangular region). We then tell ourselves that this area is a good
approximation to the area of the part of S that lies between the vertical
sides of the rectangle, and, while this is no time to get excited about the
matter, we could tell ourselves that the two areas might be exactly equal
if we choose the x shrewdly enough. The next step is to add the area of
the rectangle we have drawn to the areas of the other rectangles which we
have not drawn to obtain Mf (x) Ax. Even if we did not know in advance
that lim Ef(x) Ax exists, we should have a feeling that Ef(x) Ax should be
near ISI whenever the numbers Ax are all small (that is, whenever the
norm of P is near zero) and hence we should write
(4.481) ISI = lim Mf(x) Ax.
The final step is to recognize that the right side of this equation is the
limit of Riemann sums and hence is the Riemann integral in (4.48).
The ritual involving partitioning (or splitting up), estimating, summing
(or adding), and taking a limit to obtain a Riemann integral equal to a
number in which we are interested is known as "the process for setting up
the integral." The ability to "set up integrals" efficiently and correctly
is very valuable, and problems in calculus textbooks that require the
finding of areas are designed to promote abilities in this art. Students
cannot know, unless they are told, that they are wasting their time if they
never bother to set up integrals but only use remembered formulas to
calculate areas and volumes and the ubiquitous moments of inertia.
Problems 4.49
(^1) Figure 4.491 shows graphs of two equations y = f1(x) and y = fs(x)
which intersect at the points (-2,-6), (0,0), and (2,6). The graphs bound
lu
(lu)
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