5.2 Definite Integrals

1. 5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington

2. Quick Review

3. What you’ll learn about • Riemann Sums • The Definite Integral • Computing Definite Integrals on a Calculator • Integrability … and why The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus.

4. When we find the area under a curve by adding rectangles, the answer is called a Rieman 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.

5. 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

6. The Existence of Definite Integrals

7. 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:

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

9. Example Using the Notation

10. Example Using NINT

11. Area Under a Curve (as a Definite Integral)

12. Area • Exploration 1 p. 279

13. Riemann Sums

14. The Integral of a Constant