Q 12. Find the arc lengths of the curv... [FREE SOLUTION]

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Chapter 9: Q 12. (page 775)

Find the arc lengths of the curves defined by the parametric equations on the specified intervals.
$x = \cos h t$ , $y = t$ , $t 鈭� [0, 1]$

Short Answer

Expert verified

The arc length is $\begin{array}{rcl} \frac{1}{2} \{e - \frac{1}{e}\} \end{array}$ .

Step by step solution

Step 1. Given information.

The given parametric equations are $x = \cos h t$ and $y = t$ .

Step 2. Find the derivative of the parametric equations.

$\begin{array}{rcl} x & = & \cos h t \\ f' (t) & = & \sin h t \end{array}$ and $\begin{array}{rcl} y & = & t \\ g' (t) & = & 1 \end{array}$

Step 3. Find the length of the curve.

The length of the curve is ${鈭�}_{a}^{b} \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} d t$ , where $[a, b] = [0, 1]$ .

$\begin{array}{rcl} {鈭�}_{a}^{b} \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} d t & = & {鈭�}_{0}^{1} \sqrt{\sin^{2} h t + 1} d t \\ = & {鈭�}_{0}^{1} \sqrt{\cos^{2} h t} d t 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� \{\cos^{2} h t - \sin^{2} h t = 1\} \\ = & {鈭�}_{0}^{1} \frac{e^{t} + e^{- t}}{2} d t 鈰� 鈰� 鈰� 鈰� \{\cos h t = \frac{e^{t} + e^{- t}}{2}\} \\ = & \frac{1}{2} {鈭�}_{0}^{1} e^{t} d t + \frac{1}{2} {鈭�}_{0}^{1} e^{- t} d t \\ = & \frac{1}{2} {[e^{t}]}_{0}^{1} + \frac{1}{2} {[- e^{- t}]}_{0}^{1} \\ = & \frac{1}{2} [e^{1} - e^{0}] + \frac{1}{2} [- e^{- 1} + e^{0}] \\ = & \frac{1}{2} \{e - 1 - e^{- 1} + 1\} \\ = & \frac{1}{2} \{e - \frac{1}{e}\} \end{array}$

Step 4. Simplified answer.

Hence, the required arc length of the curve is $\begin{array}{rcl} \frac{1}{2} \{e - \frac{1}{e}\} \end{array}$ .

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91影视

Find the arc lengths of the curves defined by the parametric equations on the specified intervals.
$x = \cos h t$ , $y = t$ , $t 鈭� [0, 1]$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the parametric equations.

Step 3. Find the length of the curve.

Step 4. Simplified answer.

One App. One Place for Learning.

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91影视

Find the arc lengths of the curves defined by the parametric equations on the specified intervals.x=cosht, y=t, t鈭�0,1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the parametric equations.

Step 3. Find the length of the curve.

Step 4. Simplified answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Logic and Functions

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Find the arc lengths of the curves defined by the parametric equations on the specified intervals.
$x = \cos h t$ , $y = t$ , $t 鈭� [0, 1]$