Chapter 9: Q 12. (page 775)

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

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 $\int_{a}^{b} \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} d t$ , where $[a, b] = [0, 1]$ .

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

Step 4. Simplified answer.

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

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

Statistics

Applied Mathematics

Decision Maths

Geometry

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

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

Statistics

Applied Mathematics

Decision Maths

Geometry

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

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