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Unformatted text preview: Approximating the Definite Integral (The Area Under a Curve, Riemann Sum) Thus far we have explored the Definite Integral, b a dx ) x ( f . We will now learn how to approximate the Definite Integral (the area under the curve, Riemann Sum) which has the form, b a dx ) x ( f . The terms a and b are known as the limits of integration. We will learn how to approximate the Definite Integral (the area under the curve, Riemann Sum) with the use of rectangles and trapezoids. We will learn three methods of approximation for rectangles and one method of approximation for trapezoids. We will first begin by creating two graphs of the function 2 x 16 ) x ( f- = . Keep in mind that there is nothing special about this function. This is just the function I decided to use. The graph on the left shows the basic form of the function 2 x 16 ) x ( f- = from 5 x 5 - . Our goal will be to try to approximate the Definite Integral, ( 29 - 4 2 dx x 16 , (the area under the curve from 4 x .) 1 1 2 3 4 2.5 5 7.5 10 12.5 15 AREA f 2 x 2 2 16 2 x 2-4-2 2 4-5 5 10 15 f 2 x 2 2 16 2 x 2 Left Endpoint Approximation Method We will first learn how to approximate the area under the curve using the left endpoint method. We will begin by drawing rectangles. Notice that the approximation gets better as we draw more rectangles. Finally, notice that the blue points are all on the left hand side of the rectangles. Hence the name of the method. Lets begin approximating the definite integral using 8 rectangles. Step 1 : Split the interval [0, 4] into 8 separate intervals. The length of each interval can be determined by calculating the expression n a b- . The letters a and b come from the interval [ ] b , a . The letter n represents the number of intervals. Round to three decimal places if necessary. Notice that the number you calculated above represents the length of the base of each rectangle. Step 2 : Complete the table below. Round your answers to three decimal places. Fill in the values for n 2 1 x , , x , x , x from step 1 above. We do this by making a x = , n a b x x 1- + = , n a b x x 1 2- + = , , b x n = . Calculate the values for ( 29 ( 29 ( 29 ( 29 n 2 1 x f , , x f , x f , x f . These values will be labeled: n 2 1 y , , y , y , y . x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x ( 29 n x f y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y Notice that the values of n 2 1 y , , y , y , y represent the length of the height of each rectangle....
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