Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.1.105

Chapter 12, Problem 12.1.105

Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π

Verified step by step guidance

Recognize that the given parametric equations describe an astroid: \(x = \cos^{3} t\) and \(y = \sin^{3} t\) for \(0 \leq t \leq 2\pi\). The curve is closed and symmetric.

Recall that the area enclosed by a parametric curve \(x = x(t)\), \(y = y(t)\) for \(t\) in \([a,b]\) can be found using the formula: \(A = \int_{a}^{b} y(t) \frac{dx}{dt} \, dt\).

Compute the derivative \(\frac{dx}{dt}\): since \(x = \cos^{3} t\), use the chain rule to find \(\frac{dx}{dt} = 3 \cos^{2} t (-\sin t) = -3 \cos^{2} t \sin t\).

Set up the integral for the area: \(A = \int_{0}^{2\pi} \sin^{3} t \cdot (-3 \cos^{2} t \sin t) \, dt = -3 \int_{0}^{2\pi} \sin^{4} t \cos^{2} t \, dt\).

Use symmetry properties or trigonometric identities to simplify the integral before evaluating. For example, consider the periodicity and positivity of the integrand over \([0, 2\pi]\) or reduce the integral to a simpler form using power-reduction formulas.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. In this problem, x and y are given in terms of t, allowing the curve to be traced as t varies from 0 to 2π. Understanding how to work with parametric forms is essential for analyzing and integrating along such curves.

Area Enclosed by a Parametric Curve

The area enclosed by a parametric curve defined by x(t) and y(t) over an interval [a, b] can be found using the integral formula A = ∫ y(t) x'(t) dt. This method converts the problem of finding area into evaluating an integral involving the derivatives of the parametric functions, which is crucial for solving the given problem.

Properties of the Astroid Curve

An astroid is a specific type of hypocycloid with four cusps, often defined by x = cos³ t and y = sin³ t. Knowing its geometric properties, such as symmetry and periodicity, helps simplify calculations and understand the shape of the region whose area is to be found.

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