Explanation Given: ∫ 1 4 4 x 2 d x Formula used: Formula for the definite integral of f ( x ) from a to b : ∫ a b f ( x ) d x = lim ‖ Δ ‖ → 0 ∑ i = 1 n f ( c i ) Δ x i where, a is the lower limit of integration and b is the upper limit of integration. Formula for the sum of a constant n times: ∑ i = 1 n c = n c , where, c is a constant. Formula for the sum of first n terms: ∑ i = 1 n i = n ( n + 1 ) 2 Formula for the sum of squares of first n terms: ∑ i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 Calculation: The function f ( x ) = 4 x 2 can be integrated on the interval [ 1 , 4 ] because it is continuous on [ 1 , 4 ] . Definition of integrability implies that any partition whose norm approaches 0 can be used to determine the limit. For computational convenience, define Δ by subdividing [ 1 , 4 ] into n subintervals of equal width as below: Δ x i = 4 − ( 1 ) n = 3 n Choosing c i as the right endpoints of each subintervals produces, c i = 1 + i ( Δ x ) = 1 + i ( 3 n ) = 3 i + n n So, the definite integral is: ∫ 1 4 4 x 2 d x = lim ‖ Δ ‖ → 0 ∑ i = 1 n f ( c i ) Δ x i Since, c i = ( 3 i + n n ) lim ‖ Δ ‖ → 0 ∑ i = 1 n f ( c i ) Δ x i = lim ‖ Δ ‖ → 0 ∑ i = 1 n f ( 3 i + n n ) Δ x i Use the values of f ( 3 i + n n ) and Δ x i .So, the expression becomes, lim ‖ Δ ‖ → 0 ∑ i = 1 n f ( 3 i + n n ) Δ x i = lim n → ∞ ∑ i = 1 n 4 ( 3 i + n n ) 2 ( 3 n ) Factor out 12 n 3 from the sum as below: lim n → ∞ ∑ i = 1 n 4 ( 3 i + n n )<

11th Edition

ISBN: 9781337275347

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 4.3, Problem 8E

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To calculate: Definite integral of ∫144x2dx by the limit definition.

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Definite integral of ∫ 1 4 4 x 2 d x by the limit definition.

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