Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the four-cusped hypocycloid \(x=a \sin ^{3} t, y=a \cos ^{3} t, 0 \leq t \leq 2 \pi\), and find its length. Hint: By symmetry, you can quadruple the length of the first quadrant portion.

Step-by-Step Solution

Answer

The length of the four-cusped hypocycloid is \(3a\).

Step 1: Understand Hypocycloid Parameters

We are given the parametric equations for a four-cusped hypocycloid: \(x=a \sin^3 t\) and \(y=a \cos^3 t\). The parameter \(t\) varies from \(0\) to \(2\pi\). The parameter \(a\) is the scaling factor for the curve.

Step 2: Plotting the Curve

To sketch the hypocycloid, evaluate the parametric equations at specific values of \(t\) within \([0, 2\pi]\). First, use values like \(t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\). These give points: \((0, a), (a, 0), (0, -a), (-a, 0), (0, a)\) which outline the cusp locations. You can then sketch smooth curves connecting these points symmetrically.

Step 3: Compute Derivative for Arc Length

The length of the curve from \(t_1\) to \(t_2\) can be given by the integral \(\int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\). First, compute \(\frac{dx}{dt} = 3a \sin^2 t \cos t\) and \(\frac{dy}{dt} = -3a \cos^2 t \sin t\).

Step 4: Simplify and Integrate for Length of Curve

Substitute these derivatives into the arc length formula to get \(\int_0^{2\pi} 3a |\sin t \cos t| \, dt\). Recognize the function is symmetric; find the length from \(0\) to \(\frac{\pi}{2}\), then multiply by 4 for the complete curve using symmetry.

Step 5: Evaluate the Integral

The integral over one quadrant is \(\int_0^{\frac{\pi}{2}} 3a \sin t \cos t \, dt\). Use the identity \(\sin t \cos t = \frac{1}{2} \sin 2t\) to transform: \(\int_0^{\frac{\pi}{2}} \frac{3a}{2} \sin 2t \, dt\). The integral of \(\sin 2t\) is \(-\frac{1}{2} \cos 2t\), giving \(-\frac{3a}{4} [\cos 2t]_0^{\frac{\pi}{2}} = \frac{3a}{4} (1)\).

Step 6: Compute Total Arc Length of Hypocycloid

The length from \(0\) to \(\frac{\pi}{2}\) is \(\frac{3a}{4}\). Multiply by 4 to account for all four quadrants: final arc length is \(3a\). This is the total length of the hypocycloid.

Key Concepts

Parametric Equations

Parametric equations are an important tool in mathematics, especially for defining curves. Instead of using a single equation like \( y = f(x) \), parametric equations describe a set of related points with two separate equations, one for each coordinate, related by a third variable, often \( t \).
This third variable is called a parameter. For the hypocycloid, the parameter \( t \) allows us to smoothly draw the curve as \( t \) varies from 0 to \( 2\pi \).

• **Equation for \( x \)**: \( x = a \sin^3 t \)
• **Equation for \( y \)**: \( y = a \cos^3 t \)

By plotting these equations at different values of \( t \), you can visualize the four-cusped shape of the hypocycloid. This approach allows us to easily incorporate the cyclical nature of trigonometric functions to model the repeating, symmetric pattern of the curve.

Arc Length

Calculating the arc length of a parametric curve involves understanding the distance along the curve from one point to another. This involves a bit of calculus.
To find the total length of a curve, you essentially 'walk' along it and measure the distance you have traveled. For parametric equations, this is done with an integral that takes into account how both \( x \) and \( y \) change as \( t \) changes.The arc length \( L \) is given by: \[L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]For our hypocycloid, we need to calculate \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):

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

Substitute these derivatives into the arc length integral. The result simplifies using symmetry, allowing us to calculate the length of one segment of the curve and multiply to obtain the full length.

Curve Symmetry

The symmetry of a curve is a beautiful aspect that often simplifies calculations. Symmetry implies that the curve repeats its pattern in a regular way.
In the case of the four-cusped hypocycloid, the curve is symmetric about both the x and y axes, as well as the origin. This means each quadrant of the curve is a mirror image of the others.This property greatly simplifies the calculation of the arc length. Instead of directly computing the entire length from \( t = 0 \) to \( t = 2\pi \), we take advantage of the symmetry:

• Find the arc length from \( 0 \) to \( \pi/2 \) (first quadrant).
• Multiply the result by 4 to acquire the full length, since all quadrants are identical.

Recognizing and utilizing symmetry not only saves on computational effort but also aids in understanding the fundamental properties and beauty of mathematical curves like the hypocycloid.