Problem 34 - Calculus of a Single Variable

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 6 pi units.

Step 1: Convert from on Cartesian coordinates to polar coordinates

The equation given of the astroid is in Cartesian coordinates. It can be transformed into polar coordinates using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). This simplifies the equation to \(r = 4 \cos^{2/3}(\theta) \sin^{2/3}(\theta)\).

Step 2: Find the derivative dr/d(theta)

We will now find the derivative of \(r(\theta)\), or dr/d(theta), using the chain rule and the power rule of derivatives, which simplifies to dr/d(theta) = \(-4/3\) \([cos^{ -1/3 }(\theta)sin^{2/3 }(\theta)sin(\theta) + sin^{ -1/3 }(\theta)cos^{2/3 }(\theta)cos(\theta)]\)

Step 3: Equation for arc length

To calculate the total arc length of the graph, we use the formula for the arc length \(L = \int_a^b \sqrt{r^2 + [dr/d(\theta)]^2} d\theta \) where the limits of integration 'a' and 'b' are the values of \(\theta\) that trace out one loop of the astroid. Thus, we use \(a = 0, b = \(\pi)/2\)

Step 4: Computation of the integral

Substitute \(r\) and \(dr/d(\theta)\) from Steps 1 and 2 into the formula for arc length to get \(L = \int_0^{\(\pi)/2} \sqrt{16 cos^{4/3}(\theta)sin^{4/3}(\theta) + 16/9 [cos^2(\theta)sin^{4/3}(\theta) + sin^2(\theta)cos^{4/3 }(\theta)]^2} d\theta \) After simplification, compute the integral.

Step 5: Simplification and final computation

Upon simplification, the integral becomes \(L = 6 \int_0^{\(\pi)/2} d\theta \). Compute the integral to obtain the total length of the graph of the astroid.