To find the integral of a continuous function, one can often use another method which is much easier than taking the limit of Riemann sums. This other method is called the fundamental theorem of calculus. There are two forms to this theorem, one enabling the computation of the integral and another which gives the existence of a function whose derivative is a given function.
Proof: Let ε > 0 be given and let P be a partition a = x_{0} < x_{1} <
 (6.5) 
Then from the mean value theorem, there exists y_{i} ∈

Since ε is arbitrary, this verifies 6.4. ■
Example 6.2.2 Find ∫ _{0}^{2} cos
Note that cos
Example 6.2.3 Find ∫ _{a}^{b}αdx.
A function whose derivative is α is x → αx. Therefore, this integral is αb − αa = α
The integral ∫ _{a}^{b}f
Definition 6.2.4 Let

Observation 6.2.5 With the above definition, ∫ _{a}^{b}dx is linear satisfying 6.3 if a < b or b < a. Also ∫ _{a}^{b}αdx = αb − αa if a < b or b < a.
Note that this definition must hold if we want to continue to use Theorem 6.2.1. With this definition, one can give a convenient theorem. It holds for general Riemann integrable functions. However, I am stating it only for the case of most interest, piecewise continuous ones because I am basing the argument on Corollary 6.1.12. As noted, this corollary will end up holding in greater generality with very little change in the proof.
Theorem 6.2.6 Suppose a,b,c are all points in some interval on which f is piecewise continuous. Then
 (6.6) 
Proof: case 1: a < b < c In this case, 6.6 follows from Corollary 6.1.12.
case 2: a < c < b In this case, Corollary 6.1.12 implies

and so

case 3: c < a < b In this case, Corollary 6.1.12 implies

so
Next is the triangle inequality.
Proposition 6.2.7 Let a,b be in an interval on which f is piecewise continuous (or Riemann integrable). Then

Proof: I will give the proof for almost the only case of interest in this book, piecewise continuous. If f is piecewise continuous, then so is

The argument is the same in case a < b except you work with ∫ _{a}^{b} rather than ∫ _{b}^{a}. ■
With these basic properties of the integral, here is the other form of the fundamental theorem of calculus. This major theorem, due to Newton and Leibniz shows the existence of an “antiderivative” for any continuous function.
Proof: For t ∈

Now from Observation 6.2.5,

Therefore, by the properties of the integral given above,


Now if

Since ε is arbitrary, it follows from the definition of the limit that

Corollary 6.2.9 For F
Proof: You repeat the above argument paying attention to the sign of h. Otherwise there is no change. ■
Definition 6.2.10 When F^{′}
Proposition 6.2.11 Suppose F,G ∈∫ fdx for x in some interval. Then there exists a constant C such that F
Proof: It comes from the mean value theorem. By assumption
From Theorem 6.2.1, finding the integral of a continuous function reduces to finding an appropriate antiderivative and evaluating it at end points and subtracting. Therefore, one needs ways to find antiderivatives. Of course you can get many examples by going backwards in the tables for exponentials and logarithms and for trig. functions on Pages 360 and 354. However, usually what you have is not included directly in one of the entries of these tables. Techniques for finding antiderivatives are in the next chapter.