The Definite Integral

1. The Definite Integral In this handout: Riemann sum Definition of a definite integral Properties of the definite integral

2. |P| a = x0 x1 x2 xn-1 xn =b Riemann Sum An ordered collection P=(x0,x1,…,xn) of points of a closed interval I = [a,b] satisfying a = x0 < x1 < …< xn-1 < xn = b is a partition of the interval [a,b] into subintervals Ik=[xk-1,xk]. Let Δxk = xk-xk-1 For a partition P=(x0,x1,…,xn), let |P| = max{Δxk, k=1,…,n}. The quantity |P| is the length of the longest subinterval Ik of the partition P. Choose a sample point xi* in the subinterval [xk-1,xk]. A Riemann sum associated with a partition P and a function f is defined as:

3. Note that the integral does not depend on the choice of variable. Definition of a Definite Integral 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]. upper limit of integration Integration Symbol variable of integration (dummy variable) integrand lower limit of integration

4. Existence 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.

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

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