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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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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 11.5.10
Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.5.10Capitolo 11, Problema 11.5.10
Finding Polar Areas
Find the areas of the regions in Exercises 9–18.
Shared by the circles r = 1 and r = 2 sin θ
Guida verificata passo dopo passo1
Identify the curves given: the circles are described by the polar equations \(r = 1\) and \(r = 2 \sin \theta\).
Find the points of intersection by setting the two equations equal: \(1 = 2 \sin \theta\). Solve for \(\theta\) to determine the limits of integration.
Determine which curve lies inside and which lies outside between the points of intersection by testing values of \(\theta\) in the interval.
Set up the integral(s) for the area of the region(s) bounded by the curves. Recall that the area enclosed by a polar curve \(r(\theta)\) from \(\alpha\) to \(\beta\) is given by \(\frac{1}{2} \int_{\alpha}^{\beta} r(\theta)^2 \, d\theta\).
Express the total area as the sum or difference of integrals of \(\frac{1}{2} r^2\) for each curve over the appropriate intervals, based on which curve is outer or inner in each region.
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Polar Coordinates and Graphing
Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian coordinates. Understanding how to graph polar equations like r = 1 and r = 2 sin θ is essential to visualize the regions whose areas are to be found.
Video consigliato:
05:32
Intro to Polar Coordinates
Area Calculation in Polar Coordinates
The area enclosed by a polar curve r(θ) between angles α and β is given by the integral (1/2) ∫ from α to β of [r(θ)]² dθ. This formula is fundamental for finding areas bounded by one or more polar curves.
Video consigliato:
05:32Intro to Polar Coordinates
Finding Points of Intersection in Polar Curves
To determine the limits of integration, it is necessary to find where the polar curves intersect by solving r₁(θ) = r₂(θ). These intersection points define the boundaries of the shared region for accurate area calculation.
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Percorso guidato
09:04Slope of Polar Curves
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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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Domanda del libro di testo
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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Domanda del libro di testo
Cartesian to Polar Equations
Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
(x + 2)² + (y − 5)² = 16"
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Domanda del libro di testo
Finding Cartesian from Parametric Equations
Exercises 1–18 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. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x=√(t+1), y=√t, t ≥ 0
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Domanda del libro di testo
Hyperbolas and Eccentricity
In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.
y² − x² = 4
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