Problem 28 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \(r = 4cos^{2/3}(\theta)sin^{2/3}(\theta)\).

Step 2: 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.

Step 3: 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.