/*! 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 51 Find the arc length of the curve... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 51

Find the arc length of the curve on the interval \([0,2 \pi]\). Hypocycloid perimeter: \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\)

Short Answer

Expert verified
The arc length of the curve on the interval \([0,2 \pi]\) is 0.

Step by step solution

01

Find derivatives of x and y with respect to theta

Find the derivatives of \(x=a \cos ^{3} \theta\) and \(y=a \sin ^{3} \theta\) with respect to \(\theta\). We can use the chain rule for differentiation, which results in: \[(dx/d\theta) = -3a\cos^{2}(\theta)\sin(\theta)\] and \[(dy/d\theta) = 3a\sin^{2}(\theta)\cos(\theta)\]
02

Square the derivatives and sum them

Square each of the derivatives from the step 1 and add them. This gives us: \[g(\theta)=(dx/d\theta)^{2} +(dy/d\theta)^{2} = 9a^{2}\cos^{4}(\theta)\sin^{2}(\theta) + 9a^{2}\sin^{4}(\theta)\cos^{2}(\theta)\]
03

Simplify the expression

Now simplify the expression calculated in the previous step. We can factor out \((9a^{2}\cos^{2}(\theta)\sin^{2}(\theta))\) out of the equation obtained in step 2 to get: \[g(\theta) = 9a^{2}\cos^{2}(\theta)\sin^{2}(\theta) (\cos^{2}(\theta)+\sin^{2}(\theta))= 9a^{2}\cos^{2}(\theta)\sin^{2}(\theta)\] considering the trigonometric identity \(\cos^{2}(\theta)+\sin^{2}(\theta)=1\)
04

Find the square root of g(胃) and integrate

Take the square root of the function \(g(\theta)\) obtained in step 3 and integrate it over the given interval [0, 2蟺] using the arc length formula previously mentioned i.e., \[L = \int_{\alpha}^{\beta} \sqrt {(dx/d\theta)^{2} +(dy/d\theta)^{2}} d\theta = \int_{0}^{2\pi} \sqrt {g(\theta)}d\theta = \int_{0}^{2\pi} 3a\cos(\theta)\sin(\theta)\ d\theta\]. Save the integral part for the next step.
05

Solve the integral

The integral calculated in step 4 simplifies to 0 because \(\cos(\theta)\sin(\theta)\) is an odd function and we are integrating over the interval \([0,2\pi]\), which is symmetric about zero.

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

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \begin{aligned} &\text { Line through }\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)\\\ &x=x_{1}+t\left(x_{2}-x_{1}\right), \quad y=y_{1}+t\left(y_{2}-y_{1}\right) \end{aligned} $$

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ \begin{array}{l} x=4+2 \cos \theta \\ y=-1+\sin \theta \end{array} $$

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=a \cos \theta & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$

Give the integral formulas for the area of the surface of revolution formed when the graph of \(r=f(\theta)\) is revolved about (a) the \(x\) -axis and (b) the \(y\) -axis.

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. \(r=\frac{-6}{3+7 \sin \theta}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.