MATH 101 V2A
Integration notation
Suppose we have a function f (t) whose graph looks something like
y
t
Z
t = b, as shown.
b
f (t)dt represents the area under the graph of f (t) between t = a and
Then, the integral
a
y
t
a
b
Notice that specifying values for a and b specifies the area. That is, the area does not depend
on t.
We could also think of f as a function of the independent variable x. The graph of f (x)
will look the same as the graph of f (t) except with the t-axis replaced with an x-axis. The
Z b
integral
f (x)dx represents the area under the graph of f (x) between x = a and x = b
a
Z b
Z b
(shown below), and so
f (x)dx =
f (t)dt
a
a
y
x
a
b
We might also want to think of f as beingZa function of t, but the upper limit of integration
x
f (t)dt represents the area under the graph of
as being a variable x. Then the integral
a
f (t) between t = a and t = x (here x represents a point on the t-axis) as shown.
y
t
a
x
As you can imagine, for a fixed number a, changing the value of x results in a change in
the
Z area under the curve between t = a and t = x, and
Z therefore a change in the integral
x
x
f (t)dt. In this way, we get a function of x, F (x) =
a
f (t)dt.
a
Why do we need x? Why can’t Zwe define the same function except as a function of t?
t
Suppose we try to define G(t) =
f (t)dt. What does this mean? Does G(t) represent
a
the area under the curve from t = a to t = t (← what does that even mean?)? Let’s try
evaluating
G at a specific value ofZt. The area under the integral from t = a to t = b is given
Z
b
b
by
f (t)dt, so we want G(b) =
f (t)dt. But plugging t = b into our formula, why don’t
a
a
Z b
Z b
we get G(b) =
f (b)dt or G(b) =
f (b)db? The definition and meaning of G(t) are not
a
a
clear, which is why we need to use the variable x.
2
The Fundamental Theorem of Calculus tells us that if f (t) is continuous on an interval
containing a and x, then F 0 (x) = f (x). Notice that since F (x) only depends on x (not t),
it’s derivative should only depend on x (not t). Notice also that a small change in x results
in a smaller change in the area for those values of x for which f (x) is close to its minimum
value than those values of x for which f (x) is close to its maximum value. Somehow, this
indicates that the equation F 0 (x) = f (x) makes some sense.
y
y
t
a
t
x
a
3
x