Chapter 6: Q. 25 (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 $y_{k} = 0, 1, 2 . . . . . . n$ .
$f (x) = x^{3}$ , $[a, b] = [- 2, 2], n = 4$

Short Answer

Expert verified

The arc length is $2 (\sqrt{50} + \sqrt{2})$ .

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = x^{3}$ .

Step 2. Formula used .

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

Step 3. Find the arc length .

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

$δ x = \frac{2 + 2}{4} = 1 x_{k} = a + k \cdot δ x x_{0} = - 2 + 0 \cdot 1 = - 2, x_{1} = - 2 + 1 = - 1 x_{2} = - 2 + 2 = 0, x_{4} = - 2 + 4 = 2 δ y_{k} = (δ_{k} - δ_{k - 1}) δ y_{1} = f (x_{1}) - f (x_{0}) = - 1 + 8 = 7 δ y_{2} = f (x_{2}) - f (x_{1}) = 0 + 1 = 1 δ y_{3} = f (x_{3}) - f (x_{2}) = 1 - 0 = 1 δ y_{4} = f (x_{4}) - f (x_{3}) = 8 - 1 = 7$

Arc length $= {\sqrt{1 + (\frac{δ y_{1}}{δ x})}}^{2} \cdot δ x + \sqrt{1 + {(\frac{δ y_{2}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{3}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{4}}{δ x})}^{2}} \cdot δ x = \sqrt{1 + {(7)}^{2}} + \sqrt{2} + \sqrt{2} + \sqrt{50} = 2 \sqrt{50} + 2 \sqrt{2} = 2 (\sqrt{50} + \sqrt{2})$

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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 $y_{k} = 0, 1, 2 . . . . . . n$ .
$f (x) = x^{3}$ , $[a, b] = [- 2, 2], n = 4$

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 yk=0,1,2......n.fx=x3,a,b=-2,2,n=4

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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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 $y_{k} = 0, 1, 2 . . . . . . n$ .
$f (x) = x^{3}$ , $[a, b] = [- 2, 2], n = 4$