Explanation Given : Function y = x 4 3 and interval [ 1 , 2 ] as well as revolution of this function about x axis. There is a prebuilt formula to calculate the arc length of a given function, when the graph of a function, in a given interval is rotated along x axis and for that we need the differentiation of this function with respect to x also, so that its value can be put in said formula, to get the required arc length. Formula to be used : We have the formula that, d s = 1 + ( d y d x ) 2 d x Further arc length in the given interval [ a , b ] is calculated using the formula, Arc length = ∫ a b d s Calculation : Here for the given function y = x 4 3 , its first derivation with respect to x is : d y d x = d d x ( x 4 3 ) = 4 3 x 4 3 − 1 ( ∵ d d x x n = x n − 1 ) = 4 3 x 1 3 Thus, d s = 1 + ( 4 3 x 1 3 ) 2 d x = 1 3 9 + 16 x 2 3 d x Here we let that f ( x ) = 1 3 9 + 16 x 2 3 and g ( x ) = 1 3 9 + 16 x be two functions, so that f ( x ) ≤ g ( x ) in given interval [ 1 , 2 ] . Now we find the value of 1 3 ∫ 1 2 9 + 16 x d x and for that we let 9 + 16 x = t ∴ d d x ( 9 + 16 x ) d x = d t ⇒ 16 d x = d t ( ∵ d d x c = 0 , where c is a constant ) , 1 3 ∫ 1 2 9 + 16 x d x = 1 48

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ISBN: 9781319050733

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Chapter 9.2, Problem 34E

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To prove: that length of arc of function $y = x^{\frac{4}{3}}$ over the interval $[1, 2]$ is not less than $\frac{5}{3}$ , using comparison theorem.

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Chapter 9.2, Problem 33E

Chapter 9.2, Problem 35E

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To prove: that length of arc of function y = x 4 3 over the interval [ 1 , 2 ] is not less than 5 3 , using comparison theorem.

Solution Summary: The author explains that the length of arc of function y = x 4 3 over the interval is not less than 5 3 using comparison theorem.

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To prove: that length of arc of function $y = x^{\frac{4}{3}}$ over the interval $[1, 2]$ is not less than $\frac{5}{3}$ , using comparison theorem.

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Chapter 9 Solutions

To prove: that length of arc of function y = x 4 3 over the interval [ 1 , 2 ] is not less than 5 3 , using comparison theorem.

Solution Summary: The author explains that the length of arc of function y = x 4 3 over the interval is not less than 5 3 using comparison theorem.

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To prove: that length of arc of function y=x43 over the interval [1,2] is not less than 53, using comparison theorem.

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Chapter 9 Solutions

To prove: that length of arc of function $y = x^{\frac{4}{3}}$ over the interval $[1, 2]$ is not less than $\frac{5}{3}$ , using comparison theorem.