Introducing Approximation

# Introducing Approximation - Approximating the Definite...

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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. • Let’s 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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Introducing Approximation - Approximating the Definite...

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