The short answer: we said in class that "the total change in a
function between
t=
a and
t=
b is given by
the integral of its derivative over that interval," or something like
that. In formulas (with "int
ab" meaning the integral from
a to
b),
f(b)-f(a) = intab f'(t) dt
Remember:
the integral of a derivative gives the
change in the original function.
So, if I say that the function f(t) =
T'(t), the change in the temperature of a cup of coffee,
then the total change in the temperature in the 10 minutes starting at
time zero is the integral of this rate function:
T(10) - T(0) = int010 f(t) dt
or,
T(10) - T(0) = int010 T'(t) dt
Often we write the
antiderivative of a function using a
capital letter. So, if we did this in the coffee example above we
would say
F(
t) is the antiderivative of
f(
t) -- that is,
F(
t) is the
function we called
T(
t). The integral in this case
would be
F(10) - F(0) = int010 f(t) dt,
which is the other way we commonly write the FTC.
Where do areas come into this? Well, any definite integral is an
area -- the area between the function and the
x-axis. Thus what the FTC says is that the total change in a
function for an x interval is the area under the graph of its
derivative. If we can estimate this area (e.g., using Rieman sums, or
a calculator, Mathematica, or what have you), we can estimate
the total change in the function even if we can't find the
antiderivative of the rate (one way this could happen if we don't have
a formula for the derivative).