Domanda del libro di testo
Length in Polar Coordinates
Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.
r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2
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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 11.2.47a
Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.2.47aCapitolo 11, Problema 11.2.47a
Cycloid
a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).
Guida verificata passo dopo passo1
Identify the parametric equations of the cycloid: \(x = a(t - \sin t)\) and \(y = a(1 - \cos t)\), where \(t\) is the parameter.
Recall the formula for the arc length \(L\) of a curve defined parametrically by \(x = x(t)\) and \(y = y(t)\) over an interval \(t \in [\alpha, \beta]\):
\(L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\).
Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\):
\(\frac{dx}{dt} = a(1 - \cos t)\)
\(\frac{dy}{dt} = a \sin t\).
Substitute these derivatives into the arc length integral and simplify the integrand:
\(L = \int_{0}^{2\pi} \sqrt{a^2(1 - \cos t)^2 + a^2 \sin^2 t} \, dt\)
Simplify inside the square root to express the integrand in a simpler form.
Evaluate the integral over one full arch of the cycloid, which corresponds to \(t\) going from \(0\) to \(2\pi\), to find the length of one arch.
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For the cycloid, both x and y are given in terms of t, allowing us to analyze the curve's properties by differentiating with respect to t.
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08:02
Parameterizing Equations
Arc Length of Parametric Curves
The length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found by integrating the square root of (dx/dt)^2 + (dy/dt)^2 over that interval. This formula accounts for the combined rate of change in both x and y directions.
Video consigliato:
Percorso guidato
06:29Arc Length of Parametric Curves
Properties of the Cycloid
A cycloid is the path traced by a point on the rim of a rolling circle. One arch corresponds to t going from 0 to 2π. Understanding this interval helps set the limits for integration when calculating the length of one arch.
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Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
Polar Coordinates
Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.
a. (2, π/2)
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Domanda del libro di testo
Graphing Conic Sections
Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.
y² = −(8/3)x
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Domanda del libro di testo
Finding Parametric Equations
Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².
a. once clockwise.
(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)
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Domanda del libro di testo
Cartesian to Polar Equations
Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.
x² + y² + 5y = 0
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Domanda del libro di testo
Shifting Conic Sections
You may wish to review Section 1.2 before solving Exercises 39-56.
The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.
a. Find the center, foci, vertices, and asymptotes of the new hyperbola.
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