/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 Use the formula for the arc leng... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 61

Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Short Answer

Expert verified
The formula for the length of a polar curve is \[L=\int_a^b\sqrt{(dr/d\theta)^2+r^2}d\theta\].

Step by step solution

01

Definition of Polar Coordinates

First, remember the relation between polar and Cartesian coordinates. A point in the plane can be represented in polar coordinates as \((r,\theta)\) where \(r = \sqrt{x^2+y^2}\) and \(\theta = tan^{-1}(y/x)\). In terms of \(r\) and \(胃\), the Cartesian coordinates are expressed as \(x = r cos(胃)\) and \(y = r sin(胃)\).
02

Parametric Form Conversion

Write these relationships in parametric form, with \(\theta\) as the parameter: \(x = f(\theta) = r(\theta) cos(\theta)\), \(y = g(\theta) = r(\theta) sin(\theta)\).
03

Arc Length Formula and Derivatives

We know the arc length formula for parametric curves is defined as \[L=\int_a^b\sqrt{(dx/d\theta)^2 + (dy/d\theta)^2}d\theta\]. To use this formula to find the arc length of a polar curve, compute the derivatives \(dx/d\theta\) and \(dy/d\theta\). Using the chain rule, we get \(dx/d\theta=dr/d\theta cos(\theta)-r sin(\theta)\) and \(dy/d\theta=dr/d\theta sin(\theta)+r cos(\theta)\).
04

Substitution

Substitute these derivatives back into the arc length formula: \[L=\int_a^b\sqrt{[(dr/d\theta cos(\theta)-r sin(\theta))^2+(dr/d\theta sin(\theta)+r cos(\theta))^2]}d\theta\].
05

Expansion and Simplification

Expand and simplify the expression inside the square root: \[L=\int_a^b\sqrt{(dr/d\theta)^2+(r^2)}d\theta\].
06

Ending Point

Finally, we have derived the formula for the length of a polar curve: \[L=\int_a^b\sqrt{(dr/d\theta)^2+(r^2)}d\theta\].

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write a short paragraph describing how the graphs of curves represented by different sets of parametric equations can differ even though eliminating the parameter from each yields the same rectangular equation.

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-3}{2+4 \sin \theta}\)

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? (a) \(x=t+1, y=t^{3}\) (b) \(x=-t+1, y=(-t)^{3}\)

Use the results of Exercises \(31-34\) to find a set of parametric equations for the line or conic. $$ \text { Ellipse: vertices: }(\pm 5,0) ; \text { foci: }(\pm 4,0) $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{5}{5+3 \sin \theta}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.