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Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

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Now, lets take a trip back to Advanced Algebra. Can you find the area of the region bounded by the line x=0, y=0, y = 4 and y = 2x+3? If so, how? 3 0 4

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0 4 16

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When we find the area under a curve by adding rectangles, the answer is called a Riemann 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. Why? 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. This will be another day. We will master the Riemann Sum work first! Onwards!!!

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time velocity After 4 seconds, the object has gone 12 feet. Why? Consider 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. d’ ?

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If the velocity is not constant, we might guess that the distance traveled is still equal to the area under the curve. (The units work out.) Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. We call this the Left-hand Rectangular Approx. Method (LRAM). Approx. area:

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We could also use a Right-hand Rectangular Approximation Method(RRAM). Approx. area:

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Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). Approx area: In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

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Approximate area: width of subinterval With 8 subintervals: The exact answer for this problem is.

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Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve: What Riemann Method? Over or under estimate? Concave up or down?

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Riemann Sums Exercise Handout

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