Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 537

electric meter. You can think of the raw numeric readout on the meter as
an antiderivative of your consumption rate. The readings on the dial at the
beginning and end of the month correspond to F(a) and F(b), respectively.
Note that the raw numeric value of the reading is mostly irrelevant. It could
contain data that was influenced by somebody who lived in the house before
you. The difference between the two readings, however, is quite relevant.
It corresponds to the definite integral, and will determine how much your
electric bill is for the month.
Or consider the odometer on a car. Let's say you wanted to measure
the length of a particular journey. To do so, at the start of the trip you
would reach over to the dedicated trip odometer that every car has had
since about 1980 and press the reset button, and then at the end of the trip
you just read off the value of the trip odometer. Then you would rejoice in
not having to exercise your brain one iota or utilize a single principle from
calculus. But what if the trip odometer was broken and all you had was
the master odometer? This cannot be easily reset. 37 In this case, armed
with the calculus knowledge you gleaned from this topic (or maybe just
common sense you could have picked up anywhere), you would subtract the
odometer reading at the end of the journey from the reading at the start of
the journey to obtain the distance of the journey. The actual readings of the
odometer are F(a) and F(b), the values of the antiderivative. Just as with
the electric meter, the raw values are not useful 38 —only their difference
matters.
The first part of the fundamental theorem of calculus is very important
because it's how we actually compute integrals, at least with a pen and
paper. Remember that we defined the definite integral as a sum of a large
number of slices in the limit as the number of slices approached infinity and
the slices became infinitesimally thin. This definition is not amenable to
algebraic manipulation, like the definition of the derivative was. The first
part of the fundamental theorem of calculus says that although we may
formulate problems using the definition of the integral, we compute definite
integrals by finding an antiderivative of the function being integrated (with
pen-and-paper, at least).
The second part of the fundamental theorem of calculus is the flip side of
the first part. The first part said that definite integrals can be calculated by
using antiderivatives; the second part shows how to define an antiderivative
in terms of a definite integral.
37 Nor, as we learned from Ferris Bueller's Day Off, can it be easily rolled back.
38 At least not for this purpose. When the timing chain breaks 5 miles past your
warranty expiration, those raw values are very important.