91Ó°ÊÓ

(a) Show that the total are length of the ellipse $$ \begin{aligned} x=2 \cos t, & y=\sin t \quad(0 \leq t \leq 2 \pi) \\ \text { is given by } & \\ & 4 \int_{0}^{\pi / 2} \sqrt{1+3 \sin ^{2} t} d t \end{aligned} $$ (b) Use a CAS or a scientific calculator with a numerical integration capability to approximate the arc length in part (a). Round your answer to two decimal places. (c) Suppose that the parametric equations in part (a) describe the path of a particle moving in the \(x y\) -plane, where \(t\) is time in scientific calculator with a numerical integration capability to approximate the distance traveled by the particle from \(t=1.5\) s to \(t=4.8\) s. Round your answer to two decimal places.

Step by step solution

01

Define the arc length formula for the parametric curve

For a curve defined by the parametric equations \(x = f(t)\) and \(y = g(t)\), the arc length \(L\) from \(t = a\) to \(t = b\) can be given by the formula:\[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]

02

Compute derivatives

Find the derivatives of \(x\) and \(y\) with respect to \(t\):\[ \frac{dx}{dt} = -2\sin t \]\[ \frac{dy}{dt} = \cos t \]

03

Substitute derivatives into arc length formula

Substitute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) into the arc length formula:\[ L = \int_{0}^{2\pi} \sqrt{(-2\sin t)^2 + (\cos t)^2} \, dt \]This simplifies to:\[ L = \int_{0}^{2\pi} \sqrt{4\sin^2 t + \cos^2 t} \, dt \]

04

Simplify the integrand

Recognize the expression under the square root can be rewritten using the identity \(\sin^2 t = 1 - \cos^2 t\):\[ 4\sin^2 t + \cos^2 t = 1 + 3\sin^2 t \]Thus, the integrand becomes \(\sqrt{1 + 3\sin^2 t}\).

05

Set the limits of integration and simplify

Noticing the symmetry of the ellipse, the integral can be simplified further by evaluating from 0 to \(\pi/2\) and then multiplying the result by 4:\[ L = 4 \int_{0}^{\pi/2} \sqrt{1 + 3\sin^2 t} \, dt \]

06

Approximate arc length numerically

Use a CAS or a scientific calculator to numerically approximate the integral from 0 to \(\pi/2\):\[ \int_{0}^{\pi/2} \sqrt{1+3\sin^2 t} \, dt \approx 1.913 \]Thus, multiplying by 4, the total arc length of the ellipse is approximately:\[ 4 \times 1.913 = 7.652 \]

07

Calculate partial arc length

To find the distance traveled by the particle from \(t = 1.5\) to \(t = 4.8\), compute the integral:\[ \int_{1.5}^{4.8} \sqrt{(-2\sin t)^2 + (\cos t)^2} \, dt \]Use numerical integration to find this value:\[ \approx 3.92 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric Equations

Parametric equations offer a way to describe geometric shapes by expressing the coordinates of points on the shape as functions of an independent variable, typically noted as \(t\). This is especially useful for tracing the motion of objects. In common scenarios, \(x = f(t)\) and \(y = g(t)\) describe the position on a plane.

• Benefits: These equations allow for easy representation and analysis of curves and complex paths that would be difficult using standard methods.
• For Ellipses: The parametric form makes it simple to handle non-linear paths like ellipses, demonstrating their size, orientation, and motion over time, with just a tweak in the equations.

In our exercise, the ellipse was represented parametrically as \(x = 2 \cos t\) and \(y = \sin t\). This indicates a simple harmonic motion where \(t\) varies from \(0\) to \(2\pi\), fully tracing an ellipse in the plane.

Numerical Integration

Numerical integration allows us to approximate the value of integrals where finding an exact value analytically would be complex or impossible. It's a powerful tool in mathematics and applied science.

• Need: Sometimes, integrals have no closed-form solution, requiring approximations for practical usage.
• Methods: Common methods include the Trapezoidal Rule and Simpson’s Rule, both of which break the area under the curve into small, calculable sections.
• Technology: CAS or scientific calculators simplify numerical integration by automating these methods, giving quick, approximate answers.

In our step-by-step solution, numerical integration was crucial to estimating the arc length of the ellipse. Using a calculator, the integral was evaluated from \(0\) to \(\pi/2\), then multiplied by 4 to exploit symmetry in the ellipse, yielding an approximate arc length.

Ellipse Arc Length

Calculating the arc length of an ellipse or any parametric path involves integrating the speed, or rate of change of distance, over time along the path.

• Integration Formula: The arc length \(L\) for parametric equations is given by \( L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \).
• Symmetry Simplification: For symmetrical figures like our ellipse, the problem can sometimes be simplified by calculating a partial length and multiplying to extend across symmetrical axes.
• Ellipse Characteristics: An ellipse has a consistent, oval shape with two different axes, and its arc length requires integrating over the entire journey around the path.

In the provided solution, by finding derivatives \(\frac{dx}{dt} = -2\sin t\) and \(\frac{dy}{dt} = \cos t\), these expressed how the position changes with time. Substituting these derivatives into the arc length formula allowed for the precise computation of the path length from \(t = 1.5\) to \(t = 4.8\) seconds, using numerical evaluation for practical accuracy.