Question

Set up a definite integral that gives the arc length of the given curve. Approximate the integral using the Para. bolic Rule with \(n=8\). \(x=\sin t, y=\cos 2 t ; 0 \leq t \leq \pi / 2\)

Short answer

The arc length integral is \(L = \int_0^{\pi/2} \sqrt{\cos^2 t + 16 \sin^2 t \cos^2 t} \, dt\). Approximate this using parabolic rule with \(n=8\).

Step-by-step solution

Parametrization of Arc Length Formula

For a curve given in parametric form \((x(t), y(t))\), the arc length \(L\) from \(t=a\) to \(t=b\) is given by the integral \(L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\). Here, we have \(x=\sin t\) and \(y=\cos 2t\).

Compute the Derivatives

Find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).1. \(\frac{dx}{dt} = \cos t \)2. \(\frac{dy}{dt} = -2\sin(2t)\)

Substitute Derivatives into Arc Length Formula

Substitute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) into the arc length integral:\[L = \int_0^{\pi/2} \sqrt{(\cos t)^2 + (-2\sin 2t)^2} \, dt\].

Simplify the Integral

Simplify \((\cos t)^2 + (-2\sin 2t)^2\):- \((-2 \sin 2t)^2 = 4 \sin^2 2t = 4 (2 \sin t \cos t)^2 = 16 \sin^2 t \cos^2 t\)- So, the integral becomes: \[L = \int_0^{\pi/2} \sqrt{\cos^2 t + 16 \sin^2 t \cos^2 t} \, dt\]

Evaluate the Integral Using Parabolic Rule (n=8)

Divide the interval \([0, \pi/2]\) into 8 subintervals, each of length \(\Delta t = \frac{\pi}{16}\). Apply the trapezoidal rule approximation for each subinterval:\[L \approx \frac{\Delta t}{3} \left[f(t_0) + 4f(t_1) + 2f(t_2) + 4f(t_3) + 2f(t_4) + 4f(t_5) + 2f(t_6) + 4f(t_7) + f(t_8)\right]\]Where \(f(t) = \sqrt{\cos^2 t + 16 \sin^2 t \cos^2 t}\), and \(t_i = 0 + i \Delta t\). Evaluate \(f(t_i)\) for each \(t_i\) and substitute into the formula.

Key concepts

Understanding Parametric Equations

Parametric equations are a tool to represent a curve through equations. Instead of using a single equation in terms of x and y, we express both coordinates as functions of a parameter, often denoted as t. In our case, the parametric equations are \(x = \sin t\) and \(y = \cos 2t\). This tells us how the x and y coordinates change with the parameter t.

With parametric equations:

• You're not confined to just Cartesian equations like \(y = f(x)\).
• You can describe more complex shapes that can't be handled easily by x-y equations.
• You can easily compute derivatives that are necessary for further analysis such as finding arc lengths or tangents to the curve.

In our original problem, t changes from 0 to \(\pi/2\), describing a segment of the curve. This is significant in finding the arc length where we integrate with respect to t over this interval.

The Role of Definite Integral

The definite integral plays a pivotal role in calculating arc length for any curve defined by parametric equations. It essentially sums up an infinite number of infinitesimally small elements along the curve. The generic formula for finding the arc length \(L\) of a parametric curve \((x(t), y(t))\) from \(t = a\) to \(t = b\) is:

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

This integral formula is derived from the Pythagorean theorem by considering each infinitesimal segment of the curve as a hypotenuse of a right triangle whose legs are \(\Delta x\) and \(\Delta y\).

For our problem, the definite integral from 0 to \(\pi/2\) is:
\[L = \int_0^{\pi/2} \sqrt{\cos^2 t + 16 \sin^2 t \cos^2 t} \, dt\]

This integral, once evaluated, gives the total length of the curve between the specified limits of t. It is crucial because it turns a complex problem involving curves into a more manageable computation.

Numerical Integration Methods

Numerical integration methods are used to approximate the value of definite integrals when an analytical solution is difficult or impossible to find. In our scenario, we've utilized a method known as the Parabolic Rule, which is essentially a special case of Simpson's Rule. It approximates the area under the curve by using parabolic segments rather than just straight lines.

Here’s how it generally works:

• Subdivide the interval into an even number of subintervals; for our problem, that number is 8.
• Approximate the segments of the curve using parabolas.
• The formula for Simpson's Rule that we've used is:
  \[L \approx \frac{\Delta t}{3} \left[f(t_0) + 4f(t_1) + 2f(t_2) + 4f(t_3) + 2f(t_4) + 4f(t_5) + 2f(t_6) + 4f(t_7) + f(t_8)\right]\]
  Where \(\Delta t = \frac{\pi}{16}\) and \(f(t)\) is our function for the integrand.
• This approach generally gives a more accurate approximation than the trapezoidal rule, especially when the function is smooth.

The use of numerical methods is crucial in practical scenarios when dealing with integrals that cannot easily be solved with analytic techniques, providing an efficient and accurate method of approximation.