Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=\sin t, y=\cos t, z=\sin t ; 0 \leq t \leq 2 \pi\)

Step-by-Step Solution

Answer

The integral setup is \(\int_0^{2\pi} \sqrt{1 + \cos^2 t} \, dt\). Approximate with the Parabolic Rule for the arc length.

Step 1: Understand the Problem

We are tasked with setting up a definite integral to find the arc length of a 3D parametric curve with given equations for each coordinate in terms of the parameter \(t\).

Step 2: Recall the Arc Length Formula for Parametric Curves

For a 3D parametric curve defined by \(x(t)\), \(y(t)\), and \(z(t)\), the arc length \(L\) from \(t = a\) to \(t = b\) is given by the integral: \[ L = \int_a^b \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2 } \, dt \]

Step 3: Compute Derivatives of the Parametric Equations

Find the derivatives with respect to \(t\): \(\frac{dx}{dt} = \cos t\)\(\frac{dy}{dt} = -\sin t\)\(\frac{dz}{dt} = \cos t\)

Step 4: Set Up the Integral for Arc Length

Substitute the derivatives into the arc length formula:\[ L = \int_0^{2\pi} \sqrt{ (\cos t)^2 + (-\sin t)^2 + (\cos t)^2 } \, dt \]

Step 5: Simplify the Integral

Simplify the expression inside the square root:\[ \sqrt{ \cos^2 t + \sin^2 t + \cos^2 t } = \sqrt{ 1 + \cos^2 t } \]

Step 6: Approximate the Integral

Use the Parabolic Rule (a form of Simpson's Rule) with \(n=10\) to approximate the integral, or employ a calculator or CAS (Computer Algebra System) to evaluate it. Note that solving this integral directly requires numerical approximation.

Step 7: Optional Step: Interpretation

The curve described is a helix because the parameter \(t\) affects both \(x\) and \(z\) in a sinusoidal looping manner while the \(y\) component varies sinusoidally.

Key Concepts

3D Parametric Curves

In mathematics, a parametric curve is essentially a method of defining a curve using a parameter, typically written as functions of a variable like \( t \). When these curves extend into three dimensions, they become 3D parametric curves. These 3D curves are represented using three dependent equations: \( x(t), y(t), \) and \( z(t) \). Each equation corresponds to one spatial dimension, thus allowing the curve to traverse through a three-dimensional space.

• 3D parametric curves can trace complex shapes, from spirals to helices.
• Knowing the parameter, you can compute the exact position of any point on the curve.

In our exercise, the curve is determined by the equations \( x = \sin t \), \( y = \cos t \), and \( z = \sin t \), with \( t \) ranging from 0 to \( 2\pi \). This particular setup creates a helical structure due to the periodic sin and cos functions. Understanding how to set up and interpret these curves is critical for applications ranging from physics to computer graphics.

Definite Integral

A definite integral is a core concept of calculus that represents the accumulation of quantities, such as areas or lengths. In mathematical terms, it is expressed with upper and lower bounds, such as \( \int_a^b f(x) \, dx \), where \( a \) and \( b \) are the bounds, and \( f(x) \) is the function being integrated.

• The definite integral gives us the net area under the curve of \( f(x) \) from \( a \) to \( b \).
• It is calculated using limits of sums, which can describe physical phenomena like distance, area, and volume.

When calculating the arc length of a 3D parametric curve, we use the definite integral of the square root of the sum of the squares of the derivatives of the components, which represents the overall length of the path described by these equations.

Numerical Integration

Numerical integration involves methods for approximating the value of definite integrals. These methods are particularly useful when an exact closed-form integral is difficult to find or doesn't exist.

• It transforms complex integrals into simpler problems that a computer can solve quickly.
• Common techniques include the Trapezoidal Rule, Simpson's Rule, and others.

Given the integral we derived from the arc length formula, we needed to use numerical methods to find its value. This is because the integral involves a square root with \( \cos^2(t) \), which is challenging to integrate directly. Here, using numerical approximation allows us to get a close estimate of the arc length.

Simpson's Rule

Simpson's Rule is a popular method for numerical integration, providing an efficient way to approximate definite integrals. It is a part of the wider family of Newton-Cotes formulas and works by approximating the function with a parabolic section.

• Simpson's Rule uses quadratic polynomials to estimate the area under a curve, making it more accurate than linear methods like the trapezoidal rule.
• It's particularly effective for smooth functions and can be used with evenly spaced intervals.

In our context, we employed Simpson's Rule with \( n = 10 \) subdivisions to find the arc length. This choice helps in better handling the periodic nature of the function over one complete cycle. By dividing the integral's interval into equal parts and applying the parabolic formula, we achieve a better approximation of the true arc length.