Chapter 6: Q. 19 (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 .
$f (x) = x^{3}, [a, b] = [- 2, 2], n = 4$

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 the arc length of $f (x)$ from $x = a$ to $x = b$ is ,

Arc length of $f (x) = \lim_{n \to \infty} \sum_{k = 1}^{n} \sqrt{1 + (\frac{δ y_{k}}{δ x})} \cdot δ x$ .

where $δ x = \frac{b - a}{n}, x_{k} = a + k δ x, δ y_{k} = f (x_{k}) - f (x_{k - 1})$ for all $k = 0, 1, 2 . . . . . . . . . n$ .

Step 3. Find the arc .

$f (x) = \lim_{n \to \infty} \sum_{k = 1}^{4} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x δ x = \frac{2 + 2}{4} = 1 x_{k} = a + k δ x x_{1} = - 2 + 1 = - 1 x_{2} = - 2 + 2 = 0 x_{3} = - 2 + 3 = 1 x_{4} = - 2 + 4 = 2$

Further simplify .

$δ y_{1} = f (x_{1}) - f (x_{0}) = - 1 + 2 = 1 δ y_{2} = f (x_{2}) - f (x_{1}) = 0 - (- 1) = 1 δ y_{3} = f (x_{3}) - f (x_{2}) = 1 - 0 = 1 δ y_{4} = f (x_{4}) - f (x_{3}) = 2 - 1 = 1$

Substitute the values in formula .

localid="1649238741200" role="math" $A r c l e n g t h = \sum_{k = 1}^{4} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x = \sum_{k = 1}^{4} \sqrt{1 + {(\frac{δ y_{1}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{2}}{δ x})}^{2} \cdot} δ x + \sqrt{1 + {(\frac{δ y_{3}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{4}}{δ x})}^{2}} \cdot δ x = \sqrt{1 + {(7)}^{2}} + \sqrt{1 + 1} + \sqrt{1 + 1} + \sqrt{1 + {(7)}^{2}} = \sqrt{50} + \sqrt{2} + \sqrt{2} + \sqrt{50} = 2 (\sqrt{50} + \sqrt{2})$

Step 4. Sketch the graph of function fx=x3 .

The sketch of the given function is shown below .

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

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc .

Step 4. Sketch the graph of function fx=x3 .

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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 .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 .

Step 4. Sketch the graph of function fx=x3 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Decision Maths

Calculus

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Pure Maths

Study anywhere. Anytime. Across all devices.

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