Chapter 6: Q. 30 (page 539)

Approximate the arc length of f (x) on [a, b], using the approximation $\sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ with the given value of n. In each problem list the values of $δ y_{k}$ for $k = 0, 1, 2 . . . n$ .
f $f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 6$

Short Answer

Expert verified

The arc length is $2 \sqrt{3} + \sqrt{1} + \sqrt{- 50} + \sqrt{- 2935}$ .

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = \sqrt{9 - x^{2}}$ .

Step 2. Formula used .

The formula used to find the arc of $f (x) = \sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ .

Step 3. Find the arc length .

$f (x) = \sum_{k = 1}^{6} {\sqrt{1 + (\frac{δ y_{k}}{δ x})}}^{2} \cdot δ x$

$δ x = \frac{b - a}{n} = \frac{6}{6} = 1 x_{0} = a + k \cdot δ x = - 3 + 0 = - 3, x_{1} = - 3 + 2 = - 1 x_{2} = - 3 + 4 = 1, x_{3} = - 3 + 6 = 3 x_{4} = - 3 + 8 = 5, x_{5} = - 3 + 10 = 7 x_{5} = - 3 + 12 = 9$

$δ y_{k} = f (y_{k}) - f (y_{k - 1}) δ y_{1} = f (x_{1}) - f (x_{0}) = \sqrt{8}, δ y_{2} = f (x_{2}) - f (x_{1}) = 0 δ y_{3} = f (x_{3}) - f (x_{2}) = \sqrt{8}, δ y_{4} = f (x_{3}) - f (x_{2}) = \sqrt{- 16} δ y_{5} = f (x_{5}) - f (x_{4}) = \sqrt{- 40} - \sqrt{- 4} δ y_{6} = \sqrt{- 72} - \sqrt{- 40}$

$A r c l e n g t h = \sum_{k = 1}^{6} \sqrt{1 + {(\frac{δ y_{1}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{2}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{3}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{4}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{5}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{6}}{δ x})}^{2}} \cdot δ x = 2 \sqrt{3} + \sqrt{1} + \sqrt{1 - 51} + \sqrt{1 - 2936} = 2 \sqrt{3} + \sqrt{1} + \sqrt{- 50} + \sqrt{- 2935}$

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Approximate the arc length of f (x) on [a, b], using the approximation $\sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ with the given value of n. In each problem list the values of $δ y_{k}$ for $k = 0, 1, 2 . . . n$ .
f $f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 6$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

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Approximate the arc length of f (x) on [a, b], using the approximation ∑k=1n1+δykδx2·δxwith the given value of n. In each problem list the values of δykfor k=0,1,2...n.ffx=9-x2,a,b=-3,3,n=6

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

One App. One Place for Learning.

Most popular questions from this chapter

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Mechanics Maths

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Study anywhere. Anytime. Across all devices.

Approximate the arc length of f (x) on [a, b], using the approximation $\sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ with the given value of n. In each problem list the values of $δ y_{k}$ for $k = 0, 1, 2 . . . n$ .
f $f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 6$