33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the circle r = 8 sin θ

검증된 단계별 안내

First, recognize that the given curve is in polar coordinates: \(r = 8 \sin \theta\). This represents a circle in the polar plane.

Sketch the curve by plotting points for various values of \(\theta\) between \(0\) and \(2\pi\). Note that \(r\) is non-negative when \(\sin \theta \geq 0\), which occurs between \(0\) and \(\pi\).

Recall that the area enclosed by a polar curve \(r = f(\theta)\) from \(\theta = a\) to \(\theta = b\) is given by the formula: \[\text{Area} = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta\]

Substitute \(r = 8 \sin \theta\) into the formula to get: \[\text{Area} = \frac{1}{2} \int_{0}^{\pi} (8 \sin \theta)^2 \, d\theta = \frac{1}{2} \int_{0}^{\pi} 64 \sin^2 \theta \, d\theta\]

Simplify the integral and use the identity \(\sin^2 \theta = \frac{1 - \cos 2\theta}{2}\) to evaluate the integral over \([0, \pi]\). This will give the area of the region inside the circle.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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 an angle (r, θ). Understanding how to plot the curve r = 8 sin θ helps visualize the region bounded by the circle, which is essential for setting up the integral for the area.

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) ∫[a to b] (r(θ))^2 dθ. This formula is crucial for finding the area inside the circle r = 8 sin θ by integrating over the appropriate interval.

Determining the Limits of Integration

Identifying the correct interval for θ where the curve traces the region is necessary to compute the area accurately. For r = 8 sin θ, the curve is traced as θ goes from 0 to π, which defines the bounds for the integral.

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the circle r = 8 sin θ