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Centroids
Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.5.27
Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.5.27Capitolo 11, Problema 11.5.27
Finding Lengths of Polar Curves
Find the lengths of the curves in Exercises 21–28.
The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4
Guida verificata passo dopo passo1
Recall the formula for the length of a curve given in polar coordinates: the arc length \( L \) from \( \theta = a \) to \( \theta = b \) is given by
\[
L = \int_a^b \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta
\]
Identify the given function and interval: \( r(\theta) = \cos^3\left(\frac{\theta}{3}\right) \) and \( 0 \leq \theta \leq \frac{\pi}{4} \).
Compute the derivative \( \frac{dr}{d\theta} \) using the chain rule:
First, write \( r(\theta) = \left( \cos\left(\frac{\theta}{3}\right) \right)^3 \).
Then,
\[
\frac{dr}{d\theta} = 3 \cos^2\left(\frac{\theta}{3}\right) \cdot \left(-\sin\left(\frac{\theta}{3}\right)\right) \cdot \frac{1}{3}
\]
Simplify this expression.
Substitute \( r(\theta) \) and \( \frac{dr}{d\theta} \) into the arc length formula:
\[
L = \int_0^{\frac{\pi}{4}} \sqrt{ \left( \cos^3\left(\frac{\theta}{3}\right) \right)^2 + \left( \frac{dr}{d\theta} \right)^2 } \, d\theta
\]
Set up the integral with the simplified expressions and prepare to evaluate it either analytically (if possible) or numerically to find the length of the curve.
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Arc Length of Polar Curves
The arc length of a curve defined in polar coordinates r(θ) from θ = a to θ = b is found using the formula L = ∫ from a to b √[r(θ)² + (dr/dθ)²] dθ. This formula accounts for both the radial distance and the rate of change of r with respect to θ, providing the total length of the curve.
Video consigliato:
Percorso guidato
06:29
Arc Length of Parametric Curves
Differentiation of Polar Functions
To apply the arc length formula, you need to compute the derivative dr/dθ of the given polar function r(θ). This involves using chain and power rules, especially when r is expressed as a composite function like cos³(θ/3). Accurate differentiation is essential for evaluating the integral correctly.
Video consigliato:
05:32Intro to Polar Coordinates
Integration Techniques for Arc Length
After setting up the integral for arc length, you often need to simplify and evaluate it, which may require substitution or numerical methods if the integral is complex. Understanding how to handle trigonometric integrals and approximate definite integrals is crucial for finding the exact or approximate length.
Video consigliato:
Percorso guidato
06:29Arc Length of Parametric Curves
Pratica correlata
Domanda del libro di testo
Finding Cartesian from Parametric Equations
In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.
x = cos t, y = sin 3t
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Identifying Graphs
Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x
Then find each parabola's focus and directrix.
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Ellipses and Eccentricity
Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.
Vertices: (±10,0)
Eccentricity: 0.24
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Surface Area
Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.
x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis
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Tangent Lines to Parametrized Curves
In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.
x = sec² t − 1, y = tan t, t = −π/4
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