Chapter 6: Q. 29 (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 $k = 0, 1, 2, 3 . . . . n$ .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 3$

Short Answer

Expert verified

The arc length is $4 \sqrt{3} + \sqrt{1}$ .

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 arc length of $f (x) = \sum_{k = 1}^{3} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ .

Step 3. Find the arc length .

$f (x) = \sum_{k = 1}^{3} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$

$δ x = \frac{b - a}{n} = \frac{6}{3} = 2 x_{0} = a + k \cdot δ x = - 3, x_{1} = - 3 + 2 = - 1 x_{2} = - 3 + 4 = 1, x_{3} = - 3 + 6 = 3 δ y_{k} = f (x_{k}) - f (x_{k - 1}) δ y_{1} = f (x_{1}) - f (x_{0}) = \sqrt{8}, δ y_{2} = f (x_{2}) - f (x_{1}) = 0 δ y_{3} = f (x_{3}) - f (x_{2}) = \sqrt{8}$

$A r c l e n g t h = \sum_{k = 1}^{3} \sqrt{1 + {(\frac{δ y_{1}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{2}}{δ x})}^{2} \cdot} δ x + \sqrt{1 + {(\frac{δ y_{3}}{δ x})}^{2}} \cdot δ x = \sqrt{1 + {(\frac{\sqrt{8}}{2})}^{2}} \cdot 2 + \sqrt{1 + 0} + \sqrt{1 + {(\frac{\sqrt{8}}{2})}^{2}} \cdot 2 = 2 \sqrt{3} + \sqrt{1} + 2 \sqrt{3} = 4 \sqrt{3} + \sqrt{1}$

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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 $k = 0, 1, 2, 3 . . . . n$ .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 3$

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=0,1,2,3....n.fx=9-x2,a,b=-3,3,n=3

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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