Explanation Given: The given function is, f ( x ) = 4 x 3 The interval over which area is calculated is 2 ≤ x ≤ 5 . Formula Used: The formulas for sum of power of n natural number are, ∑ r = 1 n r 3 = n 2 ( n + 1 ) 2 4 ∑ r = 1 n r 2 = n ( n + 1 ) ( 2 n + 1 ) 6 ∑ r = 1 n C = C n Here, C is a constant. Calculation: The area A under the continuous function f ( x ) is given by the limit of the sum of the areas of approximating rectangles. Mathematically, A = lim n → ∞ ∑ r = 1 n f ( x r ) ⋅ Δ x (1) Here, Δ x is the width of an approximating rectangle x r is right side of r th rectangle. f ( x r ) is the height of the r th rectangle. To calculate area, calculate the height and width of r th rectangle. Width ( Δ x ) is b − a n . Here, b and a are end points of given interval. Use end points and calculate Δ x Δ x = 5 − 2 n Δ x = 3 n Right side of r th rectangle ( x r ) is a + r Δ x . x r = 2 + r ( 3 n ) x r = 2 + 3 r n Height of r th rectangle f ( x r ) f ( x r ) = 4 ( 2 + 3 r n ) 3 = 4 ( 8 + 27 r 3 n 3 + 3 ( 4 ) ( 3 r n ) + 3 ( 2 ) ( 9 r 2 n 2 ) ) [ ∵ ( a + b ) 3 = a 3 + b 3 + 3 a 2 b + 3 a b 2 ] = ( 32 + 108 r 3 n 3 + 144 r n + 216 r 2 n 2 ) = ( 32 + 108 r 3 n 3 + 144 r n + 216 r 2 n 2 ) Substitute ( 32 + 108 r 3 n 3 + 144 r n + 216 r 2 n 2 ) for f ( x r ) and 3 n for Δ x in equation (1) and simplify. A = lim n → ∞ ∑ r = 1 n ( 32 + 108 r 3 n 3 + 144 r n + 216 r 2 n 2 ) ⋅ ( 3 n ) = lim n → ∞ ∑ r = 1 n ( 96 n + 324 r 3 n 4 + 432 r n + 648 r 2 n 3 ) Use formulas for sum of power of n natural number and solve above equation

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ISBN: 9781305537163

Author: Stewart

Publisher: CENGAGE CO

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Chapter 13.5, Problem 18E

To determine

To Find: The area under the function f(x)=4x3 over the interval 2≤x≤5 .

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Chapter 13.5, Problem 17E

Chapter 13.5, Problem 19E

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Chapter 13 Solutions

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Ch. 13.1 - When we write limxaf(x)=L then, roughly speaking,...Ch. 13.1 - We write limxaf(x)=L and say that the ______ of...Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Ch. 13.1 - Prob. 11E Ch. 13.1 - Prob. 12E Ch. 13.1 - Prob. 13E Ch. 13.1 - Estimating Limits Numerically and Graphically Use...Ch. 13.1 - Prob. 15E Ch. 13.1 - Prob. 16E Ch. 13.1 - Prob. 17E Ch. 13.1 - Limits from a Graph For the function f whose graph...Ch. 13.1 - Limits from a Graph For the function f whose graph...Ch. 13.1 - Limits from a Graph For the function f whose graph...Ch. 13.1 - Prob. 21E Ch. 13.1 - Prob. 22E Ch. 13.1 - Prob. 23E Ch. 13.1 - Estimating Limits Graphically Use a graphing...Ch. 13.1 - Prob. 25E Ch. 13.1 - Prob. 26E Ch. 13.1 - Prob. 27E Ch. 13.1 - Prob. 28E Ch. 13.1 - Prob. 29E Ch. 13.1 - One-Sided Limits Graph the piecewise-defined...Ch. 13.1 - Prob. 31E Ch. 13.1 - Prob. 32E Ch. 13.1 - Prob. 33E Ch. 13.1 - DISCUSS: Graphing Calculator Pitfalls (a)...Ch. 13.2 - Suppose the following limits exist:...Ch. 13.2 - If f is a polynomial or a rational function and a...Ch. 13.2 - Limits from a Graph The graphs of f and g are...Ch. 13.2 - Prob. 4E Ch. 13.2 - Using Limit Laws Evaluate the limit and justify...Ch. 13.2 - Prob. 6E Ch. 13.2 - Prob. 7E Ch. 13.2 - Prob. 8E Ch. 13.2 - Prob. 9E Ch. 13.2 - Prob. 10E Ch. 13.2 - Prob. 11E Ch. 13.2 - Prob. 12E Ch. 13.2 - Prob. 13E Ch. 13.2 - Prob. 14E Ch. 13.2 - Prob. 15E Ch. 13.2 - Prob. 16E Ch. 13.2 - Prob. 17E Ch. 13.2 - Using Limit Laws Evaluate the limit and justify...Ch. 13.2 - Prob. 19E Ch. 13.2 - Prob. 20E Ch. 13.2 - Prob. 21E Ch. 13.2 - Prob. 22E Ch. 13.2 - Prob. 23E Ch. 13.2 - Prob. 24E Ch. 13.2 - Prob. 25E Ch. 13.2 - Prob. 26E Ch. 13.2 - Prob. 27E Ch. 13.2 - Prob. 28E Ch. 13.2 - Prob. 29E Ch. 13.2 - Prob. 30E Ch. 13.2 - Prob. 31E Ch. 13.2 - Prob. 32E Ch. 13.2 - Prob. 33E Ch. 13.2 - Prob. 34E Ch. 13.2 - Prob. 35E Ch. 13.2 - Prob. 36E Ch. 13.2 - Prob. 37E Ch. 13.2 - Prob. 38E Ch. 13.2 - Prob. 39E Ch. 13.2 - Prob. 40E Ch. 13.2 - Does the Limit Exist? 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The area under the function f ( x ) = 4 x 3 over the interval 2 ≤ x ≤ 5 .

Solution Summary: The author explains that the area under the function f(x)=4x 3 is given by the limit of the sum of approximating rectangles.

To Find: The area under the function f(x)=4x3 over the interval 2≤x≤5 .

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Chapter 13 Solutions