Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = sin t, y = t - cos t; 0 ≤ t ≤ π/2
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16. Parametric Equations & Polar Coordinates
Calculus with Parametric Curves
4:54 minutesProblem 12.1.105
Textbook Question
Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π
Verified step by step guidance1
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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Video duration:
4mKey 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.
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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.
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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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