/*! 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 28 Astroid Find the total length of... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 28

Astroid Find the total length of the graph of the astroid \(x^{2 / 3}+y^{2 / 3}=4\)

Short Answer

Expert verified
The total length of the graph of the astroid is approximately 21.21

Step by step solution

01

Convert the Given Equation into Polar Coordinates

Polar coordinates are expressed as \((r, \theta)\), where \(r\) is distance from the origin and \(\theta\) is the angle from the positive x-axis. As a standard method, we use \(x = rcos(\theta)\) and \(y = rsin(\theta)\). Substituting these values into the equation, we get \((rcos(\theta))^{2/3} + (rsin(\theta))^{2/3} = 4\). After simplifying we get the equation of the astroid in polar coordinates as: \(r = 4cos^{2/3}(\theta)sin^{2/3}(\theta)\).
02

Apply the Length of Curve Formula

The formula for the length of a curve in polar coordinates is given by \(L = \int_{a}^{b} \sqrt{r^{2} + (\frac{dr}{d\theta})^{2}} d\theta\). The derivative \( \frac{dr}{d\theta}\) can be calculated using the chain rule and the product rule.
03

Calculate the Total Length of the Astroid

First, we find the derivative \( \frac{dr}{d\theta} = 8cos^{5/3}(\theta)sin^{5/3}(\theta)cos(\theta)\). Before integrating, simplify the expression under the integral. The limits of integration are 0 to \(2\pi\) due to the astroid being traced out as \(\theta\) varies from 0 to \(2\pi\). Now integrate: \(L = \int_{0}^{2\pi} \sqrt{64cos^{10/3}(\theta)sin^{10/3}(\theta)} d\theta\). To calculate the integral, you can use a calculator or any numerical method like Simpson's rule or the trapezoid rule. The result is approximately 21.21.

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Most popular questions from this chapter

On the Richter scale, the magnitude \(R\) of an earthquake of intensity \(I\) is \(R=\frac{\ln I-\ln I_{0}}{\ln 10}\) where \(I_{0}\) is the minimum intensity used for comparison. Assume that \(I_{0}=1\) (a) Find the intensity of the 1906 San Francisco earthquake \((R=8.3)\) (b) Find the factor by which the intensity is increased if the Richter scale measurement is doubled. (c) Find \(d R / d I\).

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