Calculus: Analytic Geometry and Calculus, with Vectors

244 Integrals

24 If f(x) = JxJ, then f(x) = sgn x except when x = 0. When a < 0< b,
Theorem 4.37 does not guarantee correctness of the formula

f6

sgn x dx = JxJ]a

b

= jbl - lal,

but the formula may be correct anyway. What are the facts? Ans.: The

formula is correct.

25 This remark is dedicated to a distinguished professor in a distinguished

university in New Jeisey. He claimed that it does not make sense to ask a

student to evaluate the integral fo

2

x8 dx. The man was right. The integral

is a number, the limit of Riemann sums, and the number is 4. Thus, the man

was insisting that it does not make sense to ask a student to evaluate 4. What

the foxy professor really wanted to do was to emphasize the fact that fo

2

x3 dx

is something more than some black ink on white paper. It is a number. There

are times when the thing is called a symbol, but it is not a symbol. The fact

that [ foe x3 d,

I2= 16 would be hard to explain if the thing were considered

to be a symbol because we do not square symbols to get 16; we square numbers to

get 16. We must agree that we should know what we are doing when we are

asked to "evaluate" foe x3 dx and then go to work to find that the "answer" is

4. A few thoughts about these matters may even pay off sometime.

4.5 Volumes and integrals It could hardly be expected that funda-

mental ideas and definitions involving volumes of sets in E3 could be

simpler than the corresponding ideas and definitions involving areas of

sets in E. In the best treatments of the subject, the volume of a set is

its three-dimensional Lebesgue measure. The theory begins modestly

with the definition which asserts that the volume V of a rectangular

parallepiped (or brick or three-dimensional interval) having length a,

width b, and height c is the product of the dimensions, so that V = abc.

In the theory of volumes, bricks play the same role that rectangles play

in the theory of areas of sets in E2. It turns out that each bounded set

in E3 that we shall dream of considering has associated with it a number

which is the volume of the set. If two of our sets S1 and S2 are such that

S1 is a subset of S2, which means that each point of S1 is also a point of

S2, we can be sure that the volume JS1l of S1 is less than or equal to the

volume IS21 of S2. If a set S is composed of two parts S1 and S2 which

have no points in common, we can be sure that ISI = ISIJ + JS21. If one

of our sets S1 has a volume JS11 and if S2 is another set congruent to S1,

then S2 has a volume and IS21 = IS11. Appendix 2 at the end of this book

shows that the theory of volumes is (like the theory of "solid" physical

matter) not as simple as the naive believe. While a full discussion of

volumes lies far beyond the scope of this book, the theory of Lebesgue

measure in E3 justifies all of the methods we shall use for finding volumes.