## 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]*withf(x) g(x) dx = f(c) g(x) dx

### Proof

Define the numbersThen we havem = inf{ f(x): x [a, b] }

M = sup{ f(x): x [a, b] }

*m f(x) M*and since

*g*is non-negative we also have

By the properties of the Riemann integral this implies thatm g(x) f(x) g(x) M g(x)

Therefore there exists a numberm g(x) dx f(x) g(x) dx M g(x) dx

*d*between

*m*and

*M*such that

But sinced g(x) dx = f(x) g(x) dx

*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

which is what we wanted to prove.f(c) g(x) dx = f(x) g(x) dx