Q. 19 Approximate the arc length of f ... [FREE SOLUTION]

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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 $f (x) = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \sqrt{1 + (\frac{未 y_{k}}{未 x})} 路未 x$ .

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

Step 3. Find the arc .

$f (x) = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{4} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x 未 x = \frac{2 + 2}{4} = 1 x_{k} = a + k 未 x x_{1} = - 2 + 1 = - 1 x_{2} = - 2 + 2 = 0 x_{3} = - 2 + 3 = 1 x_{4} = - 2 + 4 = 2$

Further simplify .

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

Substitute the values in formula .

localid="1649238741200" role="math" $A r c l e n g t h = {鈭�}_{k = 1}^{4} \sqrt{1 + {(\frac{未 y_{k}}{未 x})}^{2}} 路未 x = {鈭�}_{k = 1}^{4} \sqrt{1 + {(\frac{未 y_{1}}{未 x})}^{2}} 路未 x + \sqrt{1 + {(\frac{未 y_{2}}{未 x})}^{2} 路} 未 x + \sqrt{1 + {(\frac{未 y_{3}}{未 x})}^{2}} 路未 x + \sqrt{1 + {(\frac{未 y_{4}}{未 x})}^{2}} 路未 x = \sqrt{1 + {(7)}^{2}} + \sqrt{1 + 1} + \sqrt{1 + 1} + \sqrt{1 + {(7)}^{2}} = \sqrt{50} + \sqrt{2} + \sqrt{2} + \sqrt{50} = 2 (\sqrt{50} + \sqrt{2})$

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

The sketch of the given function is shown below .

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91影视

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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91影视

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

Probability and Statistics

Applied Mathematics

Pure Maths

Logic and Functions

Geometry

Discrete Mathematics

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