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Chapter 6: Q. 20 (page 539)

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
$f (x) = \cos x, [a, b] = [0, 蟺], n = 3$

Short Answer

Expert verified

The arc length is $3$ .

The graph is sketched as shown below:

Step by step solution

01

Step 1. Given information .

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

02

Step 2. Formula used .

Arc length $= \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{鈭� y_{k}}{鈭� x})}^{2}} 路鈭� x$

03

Step 3. Find arc length .

$a = 0, b = 蟺, n = 3 鈭� x = \frac{b - a}{n} = \frac{蟺}{3}$

$x_{k} = a + k 鈭� x$ where k = 0,1,2,3,........n

$x_{0} = 0, x_{1} = \frac{蟺}{3}, x_{2} = \frac{2 蟺}{3}, x_{3} = \frac{3 蟺}{3}$

$鈭� y_{1} = f (x_{1}) - f (x_{0}) = \cos \frac{蟺}{3} - \cos 0 = \frac{1}{2} - 1 = - \frac{1}{2} 鈭� y_{2} = f (x_{2}) - f (x_{1}) = \cos \frac{2 蟺}{3} - \cos \frac{蟺}{3} = - \frac{1}{2} - \frac{1}{2} = - 1 鈭� y_{3} = f (x_{3}) - f (x_{2}) = \cos 蟺 - \cos \frac{2 蟺}{3} = - 1 + \frac{1}{2} = \frac{- 1}{2}$

Substitute the values in given formula ,

Arc length $= \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \sqrt{1 + {(\frac{鈭� y_{k}}{鈭� x})}^{2}} 路鈭� x$

$= [\sqrt{1 + {(- \frac{1 / 2}{蟺 / 3})}^{2}} + \sqrt{1 + {(\frac{- 1}{蟺 / 3})}^{2}} + \sqrt{1 + {(\frac{- 1 / 2}{蟺 / 3})}^{2}}] 脳 \frac{蟺}{3} = \frac{1}{6} [\sqrt{48} + \sqrt{18 路 85} + \sqrt{48}] = \frac{1}{6} [2 \sqrt{48} + \sqrt{18}] = \frac{1}{6} [2 脳 6 路 928 + 4 路 2426] = 3$

04

Step 4. Sketch of the function fx=cos x .

The sketch of the given function is shown below .

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