Explanation Approach: Let f be a continuous function on [ a , b ] and let [ a , b ] be divided into an even number n of equal subintervals by the points a = x 0 , x 1 , x 2 .... x n = b . Then by Simpson’s rule, ∫ a b f ( x ) d x = ( b − a 3 n ) { f ( x 0 ) + 4 [ f ( x 1 ) + f ( x 3 ) + f ( x 5 ) .... + f ( x n − 1 ) ] + 2 [ f ( x 2 ) + f ( x 4 ) + f ( x 6 ) .... + f ( x x − 2 ) ] + f ( x n ) } Let f be a continuous function on [ a , b ] and let [ a , b ] be divided into an even number n of equal subintervals by the points a = x 0 , x 1 , x 2 .... x n = b . Then by Trapezoidal rule, ∫ a b f ( x ) d x = ( b − a n ) { f ( x 0 ) + f ( x n ) 2 + f ( x 1 ) + ... + f ( x n − 1 ) } The Mid-point rule, uses the mid-point of each intervals as the point at which to evaluate the function for the Riemann sum. ∫ a b f ( x ) d x = b − a n ∑ a b f ( x n + x n − 1 2 ) Calculations: Consider, f ( x ) = ∫ 1 5 1 x 2 d x , Apply Trapezoidal rule, a = 1 , b = 5 and n = 8 and the end points are, x 0 = 1 , x 1 = 3 2 , x 2 = 2 , x 3 = 5 2 , x 4 = 3 , x 5 = 7 2 , x 6 = 4 , x 7 = 9 2 , x 8 = 5 i x i f ( x i ) 0 1 1 1 3 2 4 9 2 2 1 4 3 5 2 4 25 4 3 1 9 5 7 2 4 49 6 4 1 16 7 9 2 4 81 8 5 1 25 Since, b − a n = 5 − 1 8 = 1 2 Substitute into Trapezoidal rule ∫ a b f ( x ) d x = 1 2 [ 1 2 ( 1 + 1 25 ) + 4 9 + 1 4 + 4 25 + 1 9 + 4 49 + 1 16 + 4 81 ] ∫ 1 5 1 x 2 d x = ( 0.52 + 0.44 + 0.25 + 0.16 + 0.11 + 0.082 + 0.0625 + 0.0494 ) 2 = 0.837 Therefore, T = 0.837 .… ( 2 ) Apply Simpson’s rule

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Chapter 8.1, Problem 21E

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If f(x)=∫151x2dx then check whether the formula S=13[2M+T] is correct or not. Provided S is Simpson’s rule M is Midpoint rule and T is Trapezoidal rule.

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Chapter 8.1, Problem 16E

Chapter 8.1, Problem 22E

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If f ( x ) = ∫ 1 5 1 x 2 d x then check whether the formula S = 1 3 [ 2 M + T ] is correct or not. Provided S is Simpson’s rule M is Midpoint rule and T is Trapezoidal rule.

Solution Summary: The author explains how to check whether the formula S=13left[2M+Tright] is correct, where S is Simpson's rule M and T is Trapezoidal rule

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To find:

If f(x)=∫151x2dx then check whether the formula S=13[2M+T] is correct or not. Provided S is Simpson’s rule M is Midpoint rule and T is Trapezoidal rule.

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Chapter 8 Solutions

If f ( x ) = ∫ 1 5 1 x 2 d x then check whether the formula S = 1 3 [ 2 M + T ] is correct or not. Provided S is Simpson’s rule M is Midpoint rule and T is Trapezoidal rule.

Solution Summary: The author explains how to check whether the formula S=13left[2M+Tright] is correct, where S is Simpson's rule M and T is Trapezoidal rule

Concept explainers

Videos

To find: If f(x)=∫151x2dx then check whether the formula S=13[2M+T] is correct or not. Provided S is Simpson’s rule M is Midpoint rule and T is Trapezoidal rule.

Want to see the full answer?

Chapter 8 Solutions

To find:

If f(x)=∫151x2dx then check whether the formula S=13[2M+T] is correct or not. Provided S is Simpson’s rule M is Midpoint rule and T is Trapezoidal rule.