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

The region inside the lemniscate r² = 6 sin 2θ

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

45–60. Areas of regions Find the area of the following regions.The region inside the lemniscate r² = 6 sin 2θ