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4.3 Riemann Sums and Definite Integrals

4.3 Riemann Sums and Definite Integrals. HW: pg 271 1-10 (all) 12-20 (even). 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.

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4.3 Riemann Sums and Definite Integrals

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  1. 4.3 Riemann Sums and Definite Integrals HW: pg 271 1-10 (all) 12-20 (even)

  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. 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

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

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

  9. Example: Find the area between the x-axis and the curve from to . pos. neg. On the TI-89: If you use the absolute value function, you don’t need to find the roots. p

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