Definite Integrals

1. Definite Integrals 6.2

2. Summation Notation

3. When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. The width of a rectangle is called a subinterval. The entire interval is called the partition. subinterval partition Subintervals do not all have to be the same size.

4. If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by . As gets smaller, the approximation for the area gets better. subinterval partition if P is a partition of the interval

9. Riemann Sum

10. is called the definite integral of over . If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by:

11. Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

12. It is called a dummy variable because the answer does not depend on the variable chosen. upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration

13. Limit of Riemann Sum = Definite Integral

14. The Definite Integral of a Continuous Function on [a,b]

15. Existence of Definite Integrals All continuous functions are integrable.

16. We have the notation for integration, but we still need to learn how to evaluate the integral.

17. Area under a Curve

20. Example Using the Notation

21. Integral of a Constant

22. velocity time In section 5.1, we considered an object moving at a constant rate of 3 ft/sec. Since rate . time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. After 4 seconds, the object has gone 12 feet.

23. If the velocity varies: Distance: (C=0 since s=0 at t=0) After 4 seconds: The distance is still equal to the area under the curve! Notice that the area is a trapezoid.

24. Let area under the curve from a to x. (“a” is a constant) Let’s look at it another way: Then:

25. This is the definition of derivative! initial value Take the anti-derivative of both sides to find an explicit formula for area.

26. Area under curve from a to x = antiderivative at x minus antiderivative at a.

27. Area

28. Example: Find the area under the curve from x=1 to x=2. Area from x=0 to x=2 Area from x=0 to x=1

29. Area

30. The Integral of a Constant

31. Integrals on a Calculator

32. Example Using NINT

33. Example: Find the area under the curve from x=1 to x=2. To do the same problem on the TI-83: fnint(x2, x, 1, 2)

34. Example: Find the area between the x-axis and the curve from to . pos. neg.