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Theorem 7.2.8: Mean Value Theorem for Integration

If f and g are continuous functions defined on [a, b] so that g(x) 0, then there exists a number c [a, b] with
f(x) g(x) dx = f(c) g(x) dx

Proof

Define the numbers
m = inf{ f(x): x [a, b] }
M = sup{ f(x): x [a, b] }
Then we have m f(x) M and since g is non-negative we also have
m g(x) f(x) g(x) M g(x)
By the properties of the Riemann integral this implies that
m g(x) dx f(x) g(x) dx M g(x) dx
Therefore there exists a number d between m and M such that
d g(x) dx = f(x) g(x) dx
But since f is continuous on [a, b] and d is between m and M, we can apply the Intermediate Value Theorem to find a number c such that f(c) = d. Then
f(c) g(x) dx = f(x) g(x) dx
which is what we wanted to prove.
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