Riemann Sums and the Definite Integral

1. Riemann Sums and the Definite Integral

2. Why? • Why is the area of the yellow rectangle at the end = b a

3. a b Review • We partition the interval into n sub-intervals • Evaluate f(x) at right endpointsof kth sub-interval for k = 1, 2, 3, … n f(x)

4. a b Review • Sum • We expect Sn to improve thus we define A, the area under the curve, to equal the above limit. f(x) Look at Goegebra demo

5. Riemann Sum • Partition the interval [a,b] into n subintervalsa = x0 < x1 … < xn-1< xn = b • Call this partition P • The kth subinterval is xk = xk-1 – xk • Largest xk is called the norm, called ||P|| • Choose an arbitrary value from each subinterval, call it

6. Riemann Sum • Form the sumThis is the Riemann sum associated with • the function f • the given partition P • the chosen subinterval representatives • We will express a variety of quantities in terms of the Riemann sum

7. The Riemann SumCalculated • Consider the function2x2 – 7x + 5 • Use x = 0.1 • Let the = left edgeof each subinterval • Note the sum

8. The Riemann Sum • We have summed a series of boxes • If the x were smaller, we would have gotten a better approximation f(x) = 2x2 – 7x + 5

9. The Definite Integral • The definite integral is the limit of the Riemann sum • We say that f is integrable when • the number I can be approximated as accurate as needed by making ||P|| sufficiently small • f must exist on [a,b] and the Riemann sum must exist

10. Example • Try • Use summation on calculator.

11. Example • Note increased accuracy with smaller x

12. Limit of the Riemann Sum • The definite integral is the limit of the Riemann sum.

13. Properties of Definite Integral • Integral of a sum = sum of integrals • Factor out a constant • Dominance

14. Properties of Definite Integral f(x) • Subdivision rule c a b

15. Area As An Integral f(x) • The area under the curve on theinterval [a,b] A c a

16. Distance As An Integral • Given that v(t) = the velocity function with respect to time: • Then Distance traveled can be determined by a definite integral • Think of a summation for many small time slices of distance

17. Assignment • Section 5.3 • Page 314 • Problems: 3 – 47 odd