Chapter 6: Q. 24 (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 .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 6$

Short Answer

Expert verified

The arc length is $4.625$ .

The graph is sketched as below:

Step by step solution

Step 1. Given information .

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

Step 2. Formula used .

Arc length $= \lim_{n \to \infty} \sum_{k = 1}^{n} \sqrt{1 + {(\frac{\partial y_{k}}{\partial x})}^{2}} \cdot \partial x$

Step 3. Find arc length .

$f (x) = \sqrt{9 - x^{2}}, a = - 3, b = 3, n = 6 \partial x = \frac{b - a}{n} = \frac{3 + 3}{6} = 1 x_{k} = a + k \partial x x_{0} = - 3, x_{1} = - 3 + 1 = - 2 x_{2} = - 3 + 2 = - 1, x_{3} = - 3 + 3 = 0 x_{4} = - 3 + 4 = 1, x_{5} = - 3 + 5 = 2 x_{6} = - 3 + 6 = 3 \partial y_{1} = f (- 2) - f (- 3) = \sqrt{9 - 4} - \sqrt{9 - 9} = \sqrt{5} = 2.236 \partial y_{2} = f (- 1) - f (- 2) = \sqrt{9 - 1} - \sqrt{9 - 4} = \sqrt{8} - \sqrt{5} = 2.828 - 2.236 = 0.592 \partial y_{3} = f (0) - f (- 1) = \sqrt{9} - \sqrt{8} = 3 - 2.828 = 0.172 \partial y_{4} = f (1) - f (0) = \sqrt{8} - \sqrt{9} = 2.282 - 3 = - 0.172 \partial y_{5} = f (2) - f (1) = \sqrt{5} - \sqrt{8} = - 0.592 \partial y_{6} = f (3) - f (2) = 0 - \sqrt{5} = - 2.236 A r c l e n g t h = [\sqrt{1 + {(2.236)}^{2}} + \sqrt{1 + {(0.592)}^{2}} + \sqrt{1 + {(0.172)}^{2}} + \sqrt{1 + {(- 0.172)}^{2}} + \sqrt{1 + {(- 0.592)}^{2}} + \sqrt{1 + {(- 2.236)}^{2}}] \times 1 = \sqrt{5.999} + \sqrt{1.350} + \sqrt{1.0295} + \sqrt{1.0295} + \sqrt{1.350} + \sqrt{5.999} = (2.4492 + 1.1618 + 1.0146) \times 1 = 4.625$

Step 4. Sketch the graph of function fx=9-x2 .

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 .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 6$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find arc length .

Step 4. Sketch the graph of function fx=9-x2 .

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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=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 arc length .

Step 4. Sketch the graph of function fx=9-x2 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Decision Maths

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

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 .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 6$