Integral Notation
The notation for integrals is suggestive and easy to use but invites errors.
In definite integrals such as
Zb
a
f(c, y)dy
the “dy” identifies yas the variable of integration. This is a dummy variable
that gets “used up” in the integration process, so it is an illegal expression for
this variable to occur outside the integrand (the boxed area). Ry
af(c, y)dy
is the most common illegal expression, and usually indicates an error some-
where. It is always graded as wrong.
Indefinite integrals work differently: the notation g(c, y) = Rf(c, y)dy makes
sense and is an alternate notation for d
dy g(c, y) = f(c, y). Here “dy” identifies
the differentiation variable, and the final result is still a function of y.
These differences lead to some differences in techniques. For instance
when using substitution to transform an indefinite integral, a new variable is
introduced and this information must be carried along. For example
Zcos(x2) 2x dx =Zcos(u)du, where u=x2
The integral on the left is a function of x, the one on the right is a function
of u, and the “where u=x2” is necessary to relate them. The equation
is incomplete without this. This does not arise with definite integrals. See
Substitution.
The form f(b)−f(a) occurs so frequently that there is a notation for it:
f|b
a=f(b)−f(a) or f(x, c)|x=b
x=a=f(b, c)−f(a, c)
The second form is used when finvolves several symbols and it may not
be clear which is being evaluated. Using this notation the relation between
definite and indefinite integrals is written as:
Zb
a
f(x)dx = (Zf dx)|b
a
1