Problem 9 - Calculus

Question

The curve given by \(x=\cos ^{3} t, y=\sin ^{3} t\) is an astroid (ahypocycloid). Its non-parametric form is \(x^{2 / 3}+y^{2 / 3}=1\). Sketch the curve from \(t=0\) to \(t=\pi / 2\) and find its length.

Step-by-Step Solution

Answer

The length of the curve is \(\frac{3}{4}\) units.

Step 1: Parametric Equations

The curve is given by the parametric equations \(x = \cos^3 t\), \(y = \sin^3 t\). We will first analyze these parametric forms and their orientation on the axes in the first quadrant.

Step 2: Convert Parametric to Rectangular Form

First, rewrite the trigonometric identity: \(\cos^2 t + \sin^2 t = 1\). Substitute \(x = \cos^3 t\) and \(y = \sin^3 t\). We need to express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\). We find that \(\cos t = x^{1/3}\) and \(\sin t = y^{1/3}\), thus, \(x^{2/3} + y^{2/3} = 1\).

Step 3: Determine Limits of \(t\)

Given that we need to draw the curve from \(t = 0\) to \(t = \pi/2\), notice that \(x\) and \(y\) trace the astroid from the point \((1, 0)\) to \((0, 1)\) in the first quadrant.

Step 4: Sketch the Curve

Draw the curve starting at \(t = 0\): \(x = 1, y = 0\) which is point (1,0); ending at \(t = \pi/2\): \(x = 0, y = 1\), the point (0,1). The sketch results in an arc in the first quadrant of the astroid.

Step 5: Find the Length of the Curve

The formula for the length of a parametric curve \(x = f(t)\) and \(y = g(t)\) from \(t = a\) to \(t = b\) is: \[ L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \]. Find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).

Step 6: Calculate Derivatives

Calculate \( \frac{dx}{dt} = -3\cos^2 t \sin t \) and \( \frac{dy}{dt} = 3\sin^2 t \cos t \). Substitute these into the integral for the length.

Step 7: Evaluate the Integral

Substitute the derivatives into the integral: \[ L = \int_0^{\pi/2} \sqrt{9\cos^4 t \sin^2 t + 9\sin^4 t \cos^2 t} \, dt \]. Factor out common terms to simplify: \[ L = \int_0^{\pi/2} 3\cos t \sin t \, dt = 3 \int_0^{\pi/2} \frac{1}{2}\sin 2t \, dt \]. Evaluate the integral over the interval.

Step 8: Compute Simplified Integral

Further simplify and compute: \[ L = \frac{3}{2} \int_0^{\pi/2} \sin 2t \, dt = \frac{3}{2} \left[ -\frac{1}{2} \cos 2t \right]_0^{\pi/2} = \frac{3}{2} \left(0 + \frac{1}{2} \right) = \frac{3}{2} \times \frac{1}{2} = \frac{3}{4} \].

Step 9: Final Calculation

Therefore, the length of the curve from \(t = 0\) to \(t = \pi/2\) is \(\frac{3}{4}\) units.

Key Concepts

Astroid

An astroid is a particular type of hypocycloid, which is a curve generated by the path of a fixed point on a small circle rolling inside a larger circle. The classic parametric equations for an astroid are given by

• \( x = a \cos^3 t \)
• \( y = a \sin^3 t \)

For an astroid, the constant \( a \) is generally equal to one, simplifying our equations. These equations describe a star-shaped curve with four cusps. In the given exercise, the astroid is sketched from \( t = 0 \) to \( t = \pi/2 \), resulting in an arc in the first quadrant of this curve. To convert the astroid's parametric form into its non-parametric or cartesian form, we utilize trigonometric identities and transformations. Specifically,

• express \( \cos^3 t = x \)
• \( \sin^3 t = y \)
• rearrange to find \( x^{2/3} + y^{2/3} = 1 \)

This equation represents the astroid in rectangular coordinates.

Curve Length

The length of a parametric curve provides insight into how far one travels along the path of the curve between two points. For a curve described by \( x = f(t) \) and \( y = g(t) \), the formula to calculate the curve's length is:\[ L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \]Applying this to the exercise, we need to find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):

• \( \frac{dx}{dt} = -3\cos^2 t \sin t \)
• \( \frac{dy}{dt} = 3\sin^2 t \cos t \)

Inserting these into the integral allows us to determine the precise length of the curve from \( t = 0 \) to \( t = \pi/2 \). To simplify, we factor out common terms and evaluate:\[ L = \frac{3}{2} \int_0^{\pi/2} \sin 2t \, dt \]Evaluating this integral gives us \( L = \frac{3}{4} \), which is the length of the arc along the astroid in the specified interval.

Trigonometric Identities

Trigonometric identities are fundamental tools in simplifying expressions involving trigonometric functions. In this exercise, we have leveraged the Pythagorean identity:\[ \cos^2 t + \sin^2 t = 1 \]This identity is critical when transforming the astroid's parametric form to its rectangular form. By expressing \( \cos t \) and \( \sin t \) in terms of \( x \) and \( y \),

• \( \cos t = x^{1/3} \)
• \( \sin t = y^{1/3} \)

we substitute back into the Pythagorean identity to yield \( x^{2/3} + y^{2/3} = 1 \). Moreover, identities like \( \sin 2t = 2\sin t \cos t \) help simplify integral expressions, as seen when calculating the curve's length. Understanding these identities provides deeper insights into solving complex problems involving parametric equations and their transformations.

Rectangular Coordinates

Rectangular coordinates, or cartesian coordinates, involve expressing a point in a plane using an ordered pair \((x, y)\). This system is widely used to describe geometric shapes and graphs in a two-dimensional plane. In this exercise, we start with parametric equations, \( x = \cos^3 t \) and \( y = \sin^3 t \), and transform them into a rectangular form. The goal is to express the relationship as one equation, simplifying our understanding of the curve's shape. This is accomplished by finding \( x^{2/3} + y^{2/3} = 1 \), providing a clear and unified equation for the astroid in the rectangular coordinate system. This transition from parametric to rectangular forms is crucial, as it allows broader interpretation and easier manipulation of geometric shapes.