Integral Calculus, Second Mean Value Theorem
Second Mean Value Theorem
The second mean value theorem
is restricted to a closed interval,
and assumes
g and h are continuous,
with g ≥ 0.
Let l and u bound h above and below.
Because g is nonnegative,
lg ≤ gh ≤ ug throughout the interval.
Thus l times the integral of g is less than the integral of gh
is less than u times the integral of g.
Divide through by the integral of g.
Now the integral of gh over the integral of g is trapped between l and u.
By continuity there is some c in the closed interval, such that
h(c) times the integral of g gives the integral of gh.
When g is identically 0, we can't divide by the integral of g, as above,
yet the theorem still holds.
Just let c be any point between l and u.