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Chapter 7: Problem 34

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 6 pi units.

Step by step solution

01

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 \(r = 4 \cos^{2/3}(\theta) \sin^{2/3}(\theta)\).
02

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) = \(-4/3\) \([cos^{ -1/3 }(\theta)sin^{2/3 }(\theta)sin(\theta) + sin^{ -1/3 }(\theta)cos^{2/3 }(\theta)cos(\theta)]\)
03

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 = \(\pi)/2\)
04

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.
05

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.

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Most popular questions from this chapter

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