Explanation Given: y = sin x , [ 0 , π 2 ] & n=8 Key concept applied: Arc Length: We drive a formula for the length of a curve y = f ( x ) on an interval [ a , b ] . We will assume that f is continuous and differentiable on an interval [ a , b ] and we will assume that its derivative f ' is also continues on the interval [ a , b ] . We use Riemann sums to approximate the length of the curve over the interval and then take the limit to get an interval. L = ∫ a b 1 + ( d y d x ) 2 d x Midpoint: If we use the endpoints of the subintervals to approximate the integral, we run the risk that the values at the endpoints do not accurately represent the average value of the subinterval. A point which is much more likely to be close to the average would be the midpoint of each subinterval. Using the midpoint in the sum is called the midpoint rule. On the i -interval | x i − 1 , x i | we will call the midpoint x i ¯ that is Δ x [ ∑ i = 1 n f ( x i − 1 + x i 2 ) ] According to Midpoint rule: ∫ a b f ( x ) d x ≈ Δ x [ ∑ i = 1 n f ( x i − 1 + x i 2 ) ] w h e r e Δ x = ( b − a ) n Here b = Upper limit a = Lower limit n = No of interval Formula Used: ∫ a b f ( x ) d x ≈ Δ x [ ∑ i = 1 n f ( x i − 1 + x i 2 ) ] w h e r e Δ x = ( b − a ) n L = ∫ a b 1 + ( d y d x ) 2 d x Calculation: Description: y = sin x , [ 0 , π 2 ] & n=8 Step 1: Find the value of Δ x Δ x = π 2 − 0 8 = π 16 a = 0 , b = π 2 a n d n = 8 Step 2: Divide interval [ 0 , π 2 ] into n=8 subinterval of length Δ x = π 16 with the following endpoints. ∑ i = 1 8 f ( x i − 1 + x i 2 ) = f ( x 1 − 1 + x 1 2 ) + f ( x 2 − 1 + x 2 2 ) + f ( x 3 − 1 + x 3 2 ) + f ( x 4 − 1 + x 4 2 ) + f ( x 5 − 1 + x 5 2 ) + f ( x 6 − 1 + x 6 2 ) + f ( x 7 − 1 + x 7 2 ) + f ( x 8 − 1 + x 8 2 ) = f ( x 0 + x 1 2 ) + f ( x 1 + x 2 2 ) + f ( x 2 + x 3 2 ) + f ( x 3 + x 4 2 ) + f ( x 4 + x 5 2 ) + f ( x 5 + x 6 2 ) + f ( x 6 + x 7 2 ) + f ( x 7 + x 8 2 ) Sin c e x 0 = 0 , x 1 = π 16 , x 2 = π 8 a n d , x 3 = 3 π 16 , x 4 = π 4 , x 5 = 5 π 16 , x 6 = 6 π 16 , x 7 = 7 π 16 x 8 = π 2 Calculation Continued: Step 3: Now, we evaluate the function at those endpoints...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

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Chapter 9.1, Problem 12E

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The arc length of the curve over the interval using the Midpoint rule

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The arc length of the curve over the interval using the Midpoint rule

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To Find: The arc length of the curve over the interval using the Midpoint rule

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Chapter 9 Solutions

To Find:

The arc length of the curve over the interval using the Midpoint rule