Question

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

Short answer

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

Step-by-step solution

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

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.

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.