Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.3.57

Chapter 12, Problem 12.3.57

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

The region inside the lemniscate r² = 6 sin 2θ

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

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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Textbook Question

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

(2, 7π/4)

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Textbook Question

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

Textbook Question

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

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Textbook Question

27–32. Polar curves Graph the following equations.

r = 3 sin 4θ

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Textbook Question

Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.