Chapter 5 Integrals 5.2 The Definite Integral In this handout: Riemann sum Definition of a definite integral Properties of the definite integral

Riemann Sum An ordered collection P=(x 0,x 1,…,x n ) of points of a closed interval I = [a,b] satisfying a = x 0 < x 1 < …< x n-1 < x n = b is a partition of the interval [a,b] into subintervals I k =[x k-1,x k ]. Let Δx k = x k -x k-1 For a partition P=(x 0,x 1,…,x n ), let |P| = max{ Δ x k, k=1,…,n}. The quantity |P| is the length of the longest subinterval I k of the partition P. a = x 0 x1x1 x2x2 x n-1 x n =b |P||P| Choose a sample point x i * in the subinterval [x k-1,x k ]. A Riemann sum associated with a partition P and a function f is defined as:

Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) Note that the integral does not depend on the choice of variable. If f is a function defined on [a, b], the definite integral of f from a to b is the number provided that this limit exists. If it does exist, we say that f is integrable on [a, b]. Definition of a Definite Integral

Theorem: If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is integrable on [a, b]. If f is integrable on [a, b], then in calculating the value of an integral we are free to choose the partitions and sample points to simplify the calculations. It is often convenient to take a regular partition; that is, all the subintervals have the same length Δx. Existence of a Definite Integral

If the upper and lower limits are equal, then the integral is zero. Reversing the limits changes the sign. Constant multiples can be moved outside. Properties of the Integral where c is any constant Integrals can be added (or subtracted). Intervals can be added (or subtracted.)

If f(x) ≥ g(x) for a ≤ x ≤ b, then If f(x) ≥ 0 for a ≤ x ≤ b, then If m ≤ f(x) ≤ M for a ≤ x ≤ b, then Comparison Properties of the Integral