Explanation Given information: The given integral is ∫ 0 π 2 sin 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 2 sin x d x . Compare the given integral with ∫ a b f ( x ) d x . So, the function be f ( x ) = sin x . And, a = 0 , b = π 2 . Find the value of Δ x . Δ x = b − a N = π 2 − 0 6 = π 12 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 12 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 0 = f ( 0 + 0 ( π 12 ) ) = f ( 0 ) = sin 0 = 0 Substitute 1 for j , π 12 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 1 = f ( 0 + 1 ( π 12 ) ) = f ( π 12 ) = sin ( π 12 ) = 0.2588 = 0.5087 Substitute 2 for j , π 12 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 2 = f ( 0 + 2 ( π 12 ) ) = f ( 2 π 12 ) = sin ( 2 π 12 ) = 0.5 = 0.707 Substitute 3 for j , π 12 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 3 = f ( 0 + 3 ( π 12 ) ) = f ( 3 π 12 ) = sin ( 3 π 12 ) = 0.707 = 0.8408 Substitute 4 for j , π 12 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 4 = f ( 0 + 4 ( π 12 ) ) = f ( 4 π 12 ) = sin ( 4 π 12 ) = 0.866 = 0.9306 Substitute 5 for j , π 12 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 5 = f ( 0 + 5 ( π 12 ) ) = f ( 5 π 12 ) = sin ( 5 π 12 ) = 0.9659 = 0.9828 Substitute 6 for j , π 12 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 6 = f ( 0 + 6 ( π 12 ) ) = f ( 6 π 12 ) = sin ( 6 π 12 ) = 1 = 1 Now, substitute π 12 for Δ x , 0 for x 0 , 0.5087 for x 1 , 0.707 for x 2 , 0.8408 for x 3 , 0.9306 for x 4 , 0.9828 for x 5 , and 1 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 12 ( 0 + 2 ( 0.5087 ) + 2 ( 0.7071 ) + 2 ( 0.8408 ) + 2 ( 0.9306 ) + 2 ( 0.9828 ) + 1 ) = π 24 ( 1.0174 + 1.4142 + 1.6816 + 1.8612 + 1.9656 + 1 ) = π 24 ( 8.94 ) = 1.1707 Thus, the value of T 6 is 1.1707 . 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 7E

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