Problem 24 - Thomas Calculus

Question

The length of an astroid The graph of the equation \(x^{2 / 3}+\) \(y^{2 / 3}=1\) is one of a family of curves called astroids (not "asteroids") because of their star like appearance (see the accompanying figure). Find the length of this particular astroid by finding the length of half the first-quadrant portion, \(y=\left(1-x^{2 / 3}\right)^{3 / 2}\) \(\sqrt{2} / 4 \leq x \leq 1,\) and multiplying by \(8 .\)

Step-by-Step Solution

Answer

The length of the astroid is 6 units.

Step 1: Parametrize the Curve

Use the parametric equations for the astroid. For the equation \(x^{2/3} + y^{2/3} = 1\), we can write the parametric representation as \(x = a \cos^3(t)\) and \(y = a \sin^3(t)\), where \(a = 1\). So, the equations become \(x = \cos^3(t)\) and \(y = \sin^3(t)\).

Step 2: Express the Derivatives

Find the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) from the parametric equations. For \(x = \cos^3(t)\), \(\frac{dx}{dt} = -3\cos^2(t)\sin(t)\). For \(y = \sin^3(t)\), \(\frac{dy}{dt} = 3\sin^2(t)\cos(t)\).

Step 3: Set Up the Arc Length Integral

The arc length \(L\) of a parametric curve \((x(t), y(t))\) is given by the integral \( \int_{} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\). Substitute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) to get the integral \( \int_{} \sqrt{(-3\cos^2(t) \sin(t))^2 + (3\sin^2(t) \cos(t))^2} \, dt\).

Step 4: Simplify the Integral

Simplify the expression inside the square root: \[(-3\cos^2(t) \sin(t))^2 + (3\sin^2(t) \cos(t))^2 = 9(\cos^4(t) \sin^2(t) + \sin^4(t) \cos^2(t))\]Since the terms are symmetrical, we can factor: \[9(\cos^2(t) \sin^2(t)(\cos^2(t) + \sin^2(t)))\]Using \(\cos^2(t) + \sin^2(t) = 1\), this simplifies to \(9\cos^2(t)\sin^2(t)\).

Step 5: Calculate the Integral

The simplified integral now becomes:\[\int_{} \sqrt{9\cos^2(t)\sin^2(t)} \, dt = 3 \int_{} \cos(t) \sin(t) \, dt\]This integral can be solved using substitution or recognizing it as \(\frac{1}{2}\sin^2(t)\). Integrate over the appropriate bounds for half the quadrant portion \(0 \leq t \leq \frac{\pi}{4}\) for the positive quadrant. This gives:\[\frac{3}{2} \left[ \sin^2(t) \right]_{0}^{\frac{\pi}{4}} = \frac{3}{2} \left(\frac{1}{2} - 0 \right) = \frac{3}{4}\]

Step 6: Determine the Total Arc Length

Notice that the full astroid is symmetric and consists of four identical quadrants, each with two halves. Thus, multiply the result of \(\frac{3}{4}\) by 8 to get the total length:\[L = 8 \times \frac{3}{4} = 6\]

Key Concepts

Parametric equations

Parametric equations provide a way to describe a curve by defining both the x-coordinate and y-coordinate as functions of a third variable, usually denoted as \(t\). This parameter \(t\) typically represents time or another suitable variable. For our astroid, the parametric equations are given by \(x = \cos^3(t)\) and \(y = \sin^3(t)\). These equations enable us to trace the entire shape of the curve as \(t\) varies over a certain interval, in this case, from \(0\) to \(2\pi\). With parametric equations, we can easily compute properties such as derivatives and integrals, essential for finding the arc length.

Arc length integral

The arc length integral is a powerful mathematical tool to determine the length of a curve described by parametric equations. For a curve given by \((x(t), y(t))\), the arc length \(L\) is calculated using the formula:

• \[ L = \int_{} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]

This formula derives from the Pythagorean theorem and calculates the infinitesimal arc segment's length, summing these segments over the parameter interval. In our astroid example, we first need to compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\), and then substitute these into the formula to set up the integral. The next step is simplifying the expression under the square root to make evaluating the integral more manageable.

Symmetrical curves

Symmetrical curves, like the astroid, exhibit a balanced and repetitive structural feature, where the curve looks identical when rotated or reflected over specific axes. For astroids, symmetry is key because it extends uniformly in all directions, forming a star-like shape. This symmetry allows us to simplify calculations significantly. For instance, when calculating the arc length of an astroid, we only need to compute the length of half of one quadrant. Due to the astroid's symmetry, the total length is then simply this partial result multiplied by the number of symmetrical segments, which, for an astroid, happens to be 8.

Trigonometric integrals

Trigonometric integrals involve integrating functions that comprise trigonometric expressions. These are essential when working with parametric curves because the parameters often use trigonometric functions like sine and cosine. In the astroid example, after simplifying the arc length integral, we deal with the integral of the form:

• \[ \int_{} \cos(t) \sin(t) \, dt \]

Such integrals can be solved using strategies like substitution, recognizing specific identities, or directly applying known solutions. For instance, \(\cos(t)\sin(t)\) can be rewritten using the identity \(\sin(2t) = 2\sin(t)\cos(t)\), making it straightforward to integrate. Understanding these integrals and their solutions is critical for analyzing the geometric properties of curves like astroids.