Q. 26 Approximate the arc length of f ... [FREE SOLUTION]

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Chapter 6: Q. 26 (page 539)

Approximate the arc length of f (x) on [a, b], using the approximation ${鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ with the given value of n. In each problem list the values of $y_{k} = 0, 1, 2, 3 . . . . . . . . n$ .
$f (x) = \cos x, [a, b] = [0, 蟺], n = 3$

Short Answer

Expert verified

The arc length is $2 \sqrt{3601} + \sqrt{14401}$ .

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = \cos x$ .

Step 2. Formula used .

The formula to find arc length of $f (x) = {鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ .

Step 3. Find arc length .

$f (x) = {鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$

$未 x = \frac{b - a}{n} = \frac{蟺 - 0}{3} = \frac{蟺}{3} 未 k = a + k 路未 x = 0 + k \frac{蟺}{3} = \frac{蟺 k}{3} x_{0} = 0 + 0 = 0, x_{1} = 0 + 1 路 \frac{蟺}{3} = \frac{蟺}{3} x_{2} = 0 + \frac{2 蟺}{3} = \frac{2 蟺}{3}, x_{3} = 0 + \frac{3 蟺}{3} = 蟺未 y_{k} = 未_{k} - 未_{k - 1} 未 y_{1} = f (x_{1}) - f (x_{0}) = - 1 未 y_{2} = f (x_{2}) - f (x_{1}) = - 1 未 y_{3} = f (x_{3}) - f (x_{2}) = \frac{- 1}{2}$

The arc length of $f (x) = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{3} \sqrt{1 + {(\frac{未 y_{1}}{未 x})}^{2}} 路未 x + \sqrt{1 + {(\frac{未 y_{2}}{未 x})}^{2}} 路未 x + \sqrt{1 + {(\frac{未 y_{3}}{未 x})}^{2}} 路未 x = \sqrt{1 + {(\frac{- 1}{60})}^{2}} 路 60 + \sqrt{1 + {(\frac{- 1}{60})}^{2}} 路 60 + \sqrt{1 + {(\frac{- 1}{120})}^{2}} 路 60 = 2 \sqrt{3601} + \sqrt{14401}$

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91影视

Approximate the arc length of f (x) on [a, b], using the approximation ${鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ with the given value of n. In each problem list the values of $y_{k} = 0, 1, 2, 3 . . . . . . . . n$ .
$f (x) = \cos x, [a, b] = [0, 蟺], n = 3$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find arc length .

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91影视

Approximate the arc length of f (x) on [a, b], using the approximation 鈭�k=1n1+未yk未x2路未xwith the given value of n. In each problem list the values of yk=0,1,2,3........n.fx=cosx,a,b=0,蟺,n=3

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find arc length .

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Discrete Mathematics

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Geometry

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Study anywhere. Anytime. Across all devices.

Approximate the arc length of f (x) on [a, b], using the approximation ${鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x$ with the given value of n. In each problem list the values of $y_{k} = 0, 1, 2, 3 . . . . . . . . n$ .
$f (x) = \cos x, [a, b] = [0, 蟺], n = 3$