Question

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.[T] \(x=2 a \cos t-a \cos (2 t), y=2 a \sin t-a \sin (2 t), 0 \leq t<2 \pi\)

Short answer

The area enclosed by the parametric curve is \( \frac{3a^2 \pi}{2} \).

Step-by-step solution

Identify the Parametric Equations

The given parametric equations are \( x = 2a \cos t - a \cos(2t) \) and \( y = 2a \sin t - a \sin(2t) \). These describe a closed curve as \( t \) varies from 0 to \( 2\pi \).

Setup the Formula for Area Using Parametric Equations

The area \( A \) of a region bounded by parametric equations is calculated using the formula \( A = \int_{t_1}^{t_2} y \frac{dx}{dt} \, dt \). In this case, \( t_1 = 0 \) and \( t_2 = 2\pi \).

Find the Derivative \( \frac{dx}{dt} \)

Differentiate \( x \) with respect to \( t \): \( \frac{dx}{dt} = -2a \sin t + 2a \sin(2t) \).

Substitute \( y \) and \( \frac{dx}{dt} \) into the Area Integral

Substitute \( y = 2a \sin t - a \sin(2t) \) and \( \frac{dx}{dt} = -2a \sin t + 2a \sin(2t) \) into the area integral: \[ A = \int_{0}^{2\pi} (2a \sin t - a \sin(2t))(-2a \sin t + 2a \sin(2t)) \, dt \].

Simplify the Integrand

Multiply the expressions to simplify the integrand. This can be expanded and simplified to: \( A = \int_{0}^{2\pi} [ -4a^2 \sin^2 t + 4a^2 \sin t \sin(2t) + 2a^2 \sin(2t) \sin t - 2a^2 \sin^2(2t) ] \, dt \).

Use Trigonometric Identities to Simplify Further

Convert each of the trigonometric terms into simpler forms using identities such as \( \sin^2 t = \frac{1 - \cos 2t}{2} \) and \( \sin 2t = 2 \sin t \cos t \). This will help in simplifying the expression for integration.

Integrate the Simplified Expression

Integrate each term individually over the interval from 0 to \( 2\pi \). Since this is symmetric around the x-axis, many terms will cancel to zero, simplifying the result.

Calculate the Final Area

Perform the calculations for each part and sum them up to get the final result of integration. The final simplified form after integrating should yield \( A = \frac{3a^2 \pi}{2} \).

Key concepts

Area of Regions

To find the area of regions bounded by parametric curves, one works with a pair of parametric equations of the form \(x = f(t)\) and \(y = g(t)\). These equations define a curve in the plane, with each value of the parameter \(t\) corresponding to a point \((x, y)\) on the curve. The area \(A\) of a region enclosed by this parametric curve from \(t = t_1\) to \(t = t_2\) can be computed using the integral:

• \(A = \int_{t_1}^{t_2} y \frac{dx}{dt} \, dt\)

This formula arises from understanding that small changes in \(x\) (denoted as \(dx\)) and the corresponding \(y\) value create infinitesimally small strips whose area can be summed using integration.
The curve given by the parameters \(x = 2a \cos t - a \cos(2t)\) and \(y = 2a \sin t - a \sin(2t)\), describes a closed curve, and the area is obtained by evaluating the integral from \(t=0\) to \(t=2\pi\), ensuring the complete closed figure is taken into account. This approach leverages calculus to determine areas that aren't simple rectilinear shapes, providing a way to calculate more complex regions.

Integration Techniques

Integration is a fundamental technique in calculus used to find areas under curves, among other things. In our context of parametric curves, once we have separated and identified the expressions, the task is to integrate effectively.
The integrand obtained after substituting in the area formula may look intricate initially, comprising products and sums of trigonometric functions. To simplify before integrating, it helps to first multiply out any expressions, then look for terms that may cancel out during symmetry exploitation or integration.

• For example, one can expand: \((2a \sin t - a \sin(2t))(-2a \sin t + 2a \sin(2t))\)
• Subsequently, it often involves using known identities to replace complex trigonometric expressions with simpler ones.

Simplifying the integrand reduces the likelihood of computational errors and yields a manageable form for actual integration. From there, step-by-step integration of the separate components, especially noting terms cancel due to symmetry, provides a direct pathway to the area desired.

Trigonometric Identities

Trigonometric identities are invaluable tools when simplifying and integrating expressions involving trigonometric functions. They allow us to rewrite complex functions in simpler or alternative forms, making integration more accessible.
Some fundamental trigonometric identities useful in this context include:

• \(\sin^2 t = \frac{1 - \cos 2t}{2}\)
• \(\sin(2t) = 2 \sin t \cos t\)

These identities can help convert terms like \(\sin^2 t\) or \(\sin(2t)\) into forms that are more straightforward for integration or further simplification. By substituting these identities into the integral expression, you can reduce the complexity of your calculations, cancel certain terms, or expose symmetry that simplifies your problem.
Utilizing such identities effectively transforms a challenging integration task into a more mechanical process, allowing focus on calculation without unnecessary complication. With practice, recognizing when and how to apply trigonometric identities becomes an intuitive skill, aiding significantly in calculus problems involving periodic functions.