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Riemann Sum

Riemann Sum. 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.

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Riemann Sum

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  1. Riemann Sum

  2. 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.

  3. 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

  4. 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:

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

  6. Limit of Riemann Sum = Definite Integral

  7. 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

  8. The Definite Integral

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

  10. Example Using the Notation

  11. Area Under a Curve (as a Definite Integral) Note: A definite integral can be positive, negative or zero, but for a definite integral to be interpreted as an area the function MUST be continuous and nonnegative on [a, b].

  12. Area

  13. Integrals on a Calculator

  14. Example Using NINT

  15. Properties of Definite Integrals

  16. Order of Integration

  17. Zero

  18. Constant Multiple

  19. Sum and Difference

  20. Additivity

  21. Preservation of Inequality • If f is integrable and nonnegative on the closed interval [a, b], then • If f and g are integrable on the closed interval [a, b] and for every x in [a, b], then

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