Finding Polar Areas

Find the areas of the regions in Exercises 9–18.

Shared by the circles r = 1 and r = 2 sin θ

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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) 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.

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.

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.

Finding Polar AreasFind the areas of the regions in Exercises 9–18.Shared by the circles r = 1 and r = 2 sin θ