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Definite Integrals Sec. 5.2

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

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subinterval partition 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. if P is a partition of the interval

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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:

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

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

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

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

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Example: Find the area under the curve from x = 1 to x = 2. To do the same problem on the TI-89: ENTER 7 2nd

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Example: Find the area between the x-axis and the curve From to. Check the answer on the calculator! pos. neg.

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Be careful! When asked to EVALUATE AN INTEGRAL, you are finding the area between the function and the x- axis. Areas ABOVE the x-axis are positive. Areas BELOW the x-axis are negative. When asked to find the AREA of a REGION, be sure to make all negative areas positive so you’ll get the true area!

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Exploration 1 See. P. 279 of text

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Homework P. 283-284 #7-27 odd, 29-32 all, 47-56 all (DUE FRIDAY) NOTE: for problems 7-27, think about the graph of the function and finding the area geometrically!!

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