Explanation Given information: The given integral is ∫ 0 π 4 sec x d x . The value of N is 6. Formula Used: The integral is in the form of ∫ a b f ( x ) d x . The N th trapezoidal approximation is: T N = 1 2 Δ x ( x 0 + 2 x 1 + … + 2 x ( N − 1 ) + x N ) Where, Δ x = b − a N and x j = f ( a + j Δ x ) . The formula for the midpoint rule M N is: M N = Δ x ( f ( c 1 ) + f ( c 2 ) + … + f ( c N ) ) Where, Δ x = b − a N and c j = a + ( j − 1 2 ) Δ x . Calculation: The integral is I = ∫ 0 π 4 sec x d x . Compare the given integral with ∫ a b f ( x ) d x . So, the function be f ( x ) = sec x . And, a = 0 , b = π 4 . Find the value of Δ x . Δ x = b − a N = π 4 − 0 6 = π 24 To find the value of T 6 , first find the values of x 0 , x 1 , x 2 , x 3 , x 4 , x 5 and x 6 . Substitute 0 for j , π 24 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 0 = f ( 0 + 0 ( π 24 ) ) = f ( 0 ) = sec 0 = 1 Substitute 1 for j , π 24 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 1 = f ( 0 + 1 ( π 24 ) ) = f ( π 24 ) = sec ( π 24 ) = 1.0086 Substitute 2 for j , π 24 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 2 = f ( 0 + 2 ( π 24 ) ) = f ( 2 π 24 ) = sec ( 2 π 24 ) = 1.035 Substitute 3 for j , π 24 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 3 = f ( 0 + 3 ( π 24 ) ) = f ( 3 π 24 ) = sec ( 3 π 24 ) = 1.082 Substitute 4 for j , π 24 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 4 = f ( 0 + 4 ( π 24 ) ) = f ( 4 π 24 ) = sec ( 4 π 24 ) = 1.1547 Substitute 5 for j , π 24 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 5 = f ( 0 + 5 ( π 24 ) ) = f ( 5 π 24 ) = sec ( 5 π 24 ) = 1.2605 Substitute 6 for j , π 24 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 6 = f ( 0 + 6 ( π 24 ) ) = f ( 6 π 24 ) = sec ( 6 π 24 ) = 1.414 Now, substitute π 24 for Δ x , 1 for x 0 , 1.0086 for x 1 , 1.035 for x 2 , 1.082 for x 3 , 1.1547 for x 4 , 1.2605 for x 5 , and 1.414 for x 6 in the expression T N = 1 2 Δ x ( x 0 + 2 x 1 + … + 2 x ( N − 1 ) + x N ) . T 6 = 1 2 ⋅ π 24 ( 1 + 2 ( 1.0086 ) + 2 ( 1.035 ) + 2 ( 1.082 ) + 2 ( 1.1547 ) + 2 ( 1.2605 ) + 1.414 ) = π 48 ( 1 + 2.0172 + 2.07 + 2.164 + 2.3094 + 2.521 + 1.414 ) = π 48 ( 13.4956 ) = 0.883 Thus, the value of T 6 is 0.883 . Now, find the value of M N . To find the value of M 6 , first find the values of c 1 , c 2 , c 3 , c 4 , c 5 and c 6

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.9, Problem 8E

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The value of MN and TN

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Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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