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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.3.57
Capitolo 12, Problema 12.3.57

45–60. Areas of regions Find the area of the following regions.


The region inside the lemniscate r² = 6 sin 2θ

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Recognize that the given curve is a lemniscate defined by the polar equation \(r^{2} = 6 \sin 2\theta\). Our goal is to find the area enclosed by this curve.
Recall the formula for the area enclosed by a polar curve \(r = f(\theta)\) between angles \(\alpha\) and \(\beta\): \[\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r^{2} \, d\theta\]
Since the curve is symmetric, determine the appropriate interval for \(\theta\) that traces one loop of the lemniscate. For \(r^{2} = 6 \sin 2\theta\), the curve completes one loop where \(\sin 2\theta \geq 0\), typically from \(\theta = 0\) to \(\theta = \frac{\pi}{2}\). The full lemniscate consists of two such loops, so the total area will be twice the area of one loop.
Set up the integral for the area of one loop using the given equation: \[\text{Area of one loop} = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} 6 \sin 2\theta \, d\theta\]
Evaluate the integral and multiply the result by 2 to get the total area inside the lemniscate. (You can perform the integration step by step, but do not calculate the final numeric value here.)

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Polar Coordinates and Graphing

Polar coordinates represent points using a radius and angle (r, θ), which is essential for understanding curves like the lemniscate. The equation r² = 6 sin 2θ describes a curve in polar form, and interpreting this helps visualize the region whose area is to be found.
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Intro to Polar Coordinates

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) from θ = a to θ = b is given by (1/2) ∫ from a to b of [r(θ)]² dθ. This formula is crucial for finding the area inside the lemniscate, requiring integration of r² with respect to θ over the appropriate interval.
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05:32
Intro to Polar Coordinates

Properties of the Lemniscate Curve

A lemniscate is a figure-eight shaped curve defined by equations like r² = a² sin 2θ. Understanding its symmetry and periodicity helps determine the correct limits of integration and whether to multiply by a factor to account for the entire enclosed area.
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Percorso guidato
06:21
Properties of Functions
Pratica correlata
Domanda del libro di testo

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


x²/9 + y²/4 = 1

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Domanda del libro di testo

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

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Domanda del libro di testo

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The upper half of the parabola x=y ², originating at (0, 0)

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Domanda del libro di testo

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(2, 7π/4)

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Domanda del libro di testo

What is the polar equation of a circle of radius √(a²+b²) centered at (a, b)?

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Domanda del libro di testo

75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

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