1. CHAPTER 2 2.4 Continuity The Definite Integral animation  i=1 n f (x i * )  x f (x) xx Riemann Sum xi*xi* xixi x i+1

2. Definition of a Definite Integral If f is a continuous function defined for b  x  a we divide the interval [a,b] into subintervals of equal width  x =(b–a)/n. We let x 0 (= a), x 1, x 2 … x n ( = b) be the endpoints of these subintervals and we choose sample points x 1 *, x 2 * … x n *, so x i * lies in the ith subinterval [x i-1, x i ]. Then the definite integral of f from a to b is b  a b f (x) dx = lim n  0  i=1 n f (x i * )  x.

3. Example If f (x) = x 2, 0 <= x <= 1, evaluate the Riemann sum with n = 4, taking the sample points to be right endpoints. Right endpoints animation

4. Example If f (x) = x 2, 0 <= x <= 1, evaluate the Riemann sum with n = 4, taking the sample points to be left endpoints. Left endpoints animation

5. Example If f (x) = x 2, 0 <= x <= 1, evaluate the Riemann sum with n = 4, taking the sample points to be midpoints. Midpoints animation

6. Midpoint Rule  b a f (x) dx   n i = 1 f (x i * )  x =  x [ f (x 1 ) + … + f (x n )] where  x = (b – a) / n and x i = ½ ( x i-1 + x i ) = midpoint of [x i-1, x i ]. _ _ _

7. Properties of the Integral 1.  a b c dx = c(b – a), where c is any constant. 2.  a b [ f (x) + g(x)]dx =  a b f (x) dx +  a b g(x) dx 3.  a b c f (x) dx = c  a b f (x) dx, 4.  a b [ f (x) - g(x)]dx =  a b f (x) dx -  a b g(x) dx. 6.  a b f (x) dx +  a b f (x) dx =  a b f(x) dx.

8. Example Solve :  -1 3 |3x – 5| dx.