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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.PE.17
Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.PE.17Capitolo 11, Problema 11.PE.17
Lengths of Curves
Find the lengths of the curves in Exercises 13–19.
x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2
Guida verificata passo dopo passo1
Recall the formula for the length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\) is:
\[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Identify the given functions:
\[x(t) = 5 \cos t - \cos 5t\]
\[y(t) = 5 \sin t - \sin 5t\]
and the interval is \(0 \leq t \leq \frac{\pi}{2}\).
Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\):
For \(x(t)\), use the derivative of cosine:
\[\frac{dx}{dt} = -5 \sin t + 5 \sin 5t\]
For \(y(t)\), use the derivative of sine:
\[\frac{dy}{dt} = 5 \cos t - 5 \cos 5t\]
Substitute these derivatives into the arc length formula under the square root:
\[L = \int_0^{\frac{\pi}{2}} \sqrt{\left(-5 \sin t + 5 \sin 5t\right)^2 + \left(5 \cos t - 5 \cos 5t\right)^2} \, dt\]
Simplify the expression inside the square root if possible, then set up the integral for evaluation. This integral may require numerical methods or further trigonometric simplifications to solve.
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with parametric forms is essential for calculating properties like curve length.
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08:02
Parameterizing Equations
Arc Length Formula for 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 the sum of the squares of the derivatives: ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt. This formula generalizes the Pythagorean theorem to infinitesimal segments along the curve.
Video consigliato:
Percorso guidato
06:29Arc Length of Parametric Curves
Differentiation of Trigonometric Functions
Calculating the derivatives dx/dt and dy/dt requires knowledge of differentiating trigonometric functions like sine and cosine. Recognizing the derivatives (d/dt) sin t = cos t and (d/dt) cos t = -sin t is crucial for correctly applying the arc length formula to the given parametric equations.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
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Identifying Parametric Equations in the Plane
Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.
x = √t, y = 1 − √t, t ≥ 0
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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
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
Area in Polar Coordinates
Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.
Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ
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Domanda del libro di testo
Polar to Cartesian Equations
Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.
r cos (θ − 3π/4) = (√2)/2
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