## 7.1. Riemann Integral

### Corollary 7.1.20: Integral Evaluation Shortcut

Suppose

*f*is an continuous function defined on the closed, bounded interval*[a, b]*, and*G*is a function on*[a, b]*such that*G'(x) = f(x)*for all*x*in*(a, b)*. Thenf(x) dx = G(b) - G(a)

**Note:**The function*G*is often called**Antiderivative**of*f*, and this corollary is called**First Fundamental Theorem of Calculus**in many calculus text books. Those books also define a Second Fundamental Theorem of Calculus, which*we*called*the*Fundamental Theorem of Calculus.### Proof:

This being a corollary means that it must be easy to prove. We already know from the previous theorem that if we define the functionthenF(x) = f(t) dt

*F(b) - F(a) = f(x) dx*and

*F' = f*. What we need to prove is that if we take

*any*function

*G*such that

*G'(x) = f(x)*then

*G(b) - G(a) = f(x) dx*also. So, define

whereH(x) = F(x) - G(x)

*F*and

*G*are as defined above. Then

so thatH'(x) = F'(x) - G'(x) = f(x) - f(x) = 0

*H(x) = c*for some constant

*c*. But then

*F(x) = G(x) + c*so that

F(b) - F(a) = (G(a) + c) - (G(b) + c) = G(b) - G(a)