Chapter 9: Q 69. (page 723)

Use your result from Exercise $68$ to show that the arc length formula for a function $y = f (x)$ is a special case of the arc length formula for a parametric curve.

Short Answer

Expert verified

$\int_{a}^{b} \sqrt{1 + {(f^{'} (t))}^{2}} d t$

Step by step solution

Given information

$y = f (x)$

Calculation

Consider the function $y = f (x)$

For a parameter $t$ the function $y = f (t)$ for some $t \in [a, b]$

The goal is to determine the curve's arc length.

If the curve $C$ is expressed by parametric equations $x = f (t), y = y (t)$ on the interval $[a, b]$ then the arc length is given by the formula,

$\int_{a}^{b} \sqrt{{(f^{'} (t))}^{2} + {(g^{'} (t))}^{2}} d t$

Thus, $f (t) = x \Rightarrow f^{'} (t) = \frac{d x}{d t}$

$g (t) = y \Rightarrow g^{'} (t) = \frac{d y}{d t}$

Substituting the values of $f^{'} (t), g^{'} (t)$ then the arc length is

Arc length $= \int_{a}^{b} \sqrt{{(\frac{d}{d t} x)}^{2} + {(\frac{d}{d t} y)}^{2}} d t$

$= \int_{a}^{b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$

Then,

Arc length $= \int_{a}^{b} \sqrt{1 + {(\frac{d y}{d t})}^{2}} d t$

Arc length $= \int_{a}^{b} \sqrt{1 + {(f^{'} (t))}^{2}} d t [$ since , for some parameter $t]$ Therefore the arc length of the curve $y = f (t)$ is $\int_{a}^{b} \sqrt{1 + {(f^{'} (t))}^{2}} d t$

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Use your result from Exercise $68$ to show that the arc length formula for a function $y = f (x)$ is a special case of the arc length formula for a parametric curve.

Short Answer

Step by step solution

Given information

Calculation

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Use your result from Exercise 68 to show that the arc length formula for a function y=f(x) is a special case of the arc length formula for a parametric curve.

Short Answer

Step by step solution

Given information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Probability and Statistics

Mechanics Maths

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Use your result from Exercise $68$ to show that the arc length formula for a function $y = f (x)$ is a special case of the arc length formula for a parametric curve.