Explanation Formula used: Trapezoidal rule: Let f be a function on an interval [ a , b ] . Partition [ a , b ] into n subintervals of equal length Δ x = ( b − a ) / n with end points a = x 0 < x 1 < x 2 < ... < x n = b . Then T n = [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + .... + 2 f ( x n − 1 ) + f ( x n ) ] Δ x / 2 is an approximation of ∫ a b f ( x ) d x . Calculation: The given integral is ∫ 1 11 x 3 d x . So, the function is f ( x ) = x 3 . The given interval is [ a , b ] = [ 1 , 11 ] and number of partitions n = 5 . Partition the interval [ 1 , 11 ] into 5 equal subintervals. The length of the each interval is Δ x = ( b − a ) / n . Substitute a = 1 , b = 11 and n = 5 . Δ x = ( 11 − 1 ) 5 = 10 5 = 2 Thus, the width of each interval is 2. So, the end points are x 0 = 1 , x 1 = 3 , x 2 = 5 , x 3 = 7 , x 4 = 9 and x 5 = 11 . Calculate the value of the function f ( x ) = x 3 at each end point as follows. x f ( x ) 1 x 3 = 1 3 = 1 3 x 3 = 3 3 = 27 5 x 3 = 5 3 = 125 7 x 3 = 7 3 = 343 9 x 3 = 9 3 = 729 11 x 3 = 11 3 = 1331 Table 1 By trapezoidal rule, T n = [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) +

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Chapter 6.4, Problem 30E

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The approximate value of the integral ∫111x3dx using the trapezoidal rule and to find the exact value of the integral ∫111x3dx using fundamental theorem of calculus.

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Chapter 6.4, Problem 29E

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Calculus for Business Economics Life Sciences and Social Sciences Plus NEW

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The approximate value of the integral ∫ 1 11 x 3 d x using the trapezoidal rule and to find the exact value of the integral ∫ 1 11 x 3 d x using fundamental theorem of calculus.

Solution Summary: The author calculates the approximate value of the integral using the trapezoidal rule and the fundamental theorem of calculus.

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The approximate value of the integral ∫111x3dx using the trapezoidal rule and to find the exact value of the integral ∫111x3dx using fundamental theorem of calculus.

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