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"Fundamental" genre: Noise

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Calculus is fundamental.

Lyrics
the theorem:

if

little f of t is continuous on the open interval I
and
a is a constant on I
and
x and t are in I
and
big F of t is any antiderivative of little f of t,

then

the integral of little f of t
with respect to t
from a to x
is equal to
big F of x

the proof:

define big A of x to be
the integral of little f of t
with respect to t
from a to x

thus d big A dx
is equal to
the limit as h goes to naught of
big A of quantity x plus h less big A of x
all over h

for any h greater than naught
such that x plus h is in I
big A of quantity x plus h
is equal to
the integral of little f of t
with respect to t
from a to quantity x plus h

thus big A of quantity x plus h less big A of x
is equal to
the integral of little f of t
with respect to t
from a to quantity x plus h
less the integral of little f of t
with respect to t
from a to x

which is equal to
the integral of little f of t
with respect to t
from x to a
plus the integral of little f of t
with respect to t
from a to quantity x plus h

[note that an integral
from r to s
is equal to
the same integral
from s to r
multiplied by negative one]

which is equal to
the integral of little f of t
with respect to t
from x to quantity x plus h

[note that an integral
from b to c
added to the same integral
from c to d
is equal to
the same integral
from b to d]

let little m be the minimum value
of little f of t
on the interval
from x to quantity x plus h

let big M be the maximum value
of little f of t
on the interval
from x to quantity x plus h

little m multiplied by h
must be less than (or equal to)
the integral of little f of t
with respect to t
from x to quantity x plus h
which must be less than (or equal to)
big M multiplied by h

if we divide the inequality through by h
we see that
little m
is less than (or equal to)
the integral of little f of t
with respect to t
from x to quantity x plus h
all over h
which is less than (or equal to)
big M

this inequality is the same as
little m
is less than (or equal to)
big A of quantity x plus h
less big A of x
all over h
which is less than (or equal to)
big M

take the limit as h goes to naught

little m
goes to little f of x

big A of quantity x plus h
less big A of x
all over h
goes to d big A dx

big M
goes to little f of x

[remember that
little m
and big M
were defined as
the minimum
and maximum
of little f of t
on the interval
from x to quantity x plus h
respectively

as h goes to naught
the minimum
and maximum
go towards little f of x]

thus little f of x
is less than (or equal to)
d big A dx
which is less than (or equal to)
little f of x

or
little f of x
equals
d big A dx

or
the antiderivative of little f of x
is big A of x

or
the integral of little f of t
with respect to t
from a to x
is equal to
big F of x

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