Question

45–60. Areas of regions Find the area of the following regions.The region inside the rose r = 4 sin 2θ and inside the circle r = 2

Accepted Answer

First, understand the problem: we need to find the area of the region that lies both inside the rose curve given by $$r = 4 \sin 2	heta$$ and inside the circle given by $$r = 2. $$This means we are looking for the overlapping area between these two curves. Next, find the points of intersection between the two curves by setting their $$r$$ values equal: $$4 \sin 2	heta = 2. $$Solve for $$	heta to $$determine the limits of integration for the overlapping region. Recall that the area enclosed by a polar curve $$r(	heta)$$ between angles $$\alpha$$ and $$\beta is $$given by the integral $$\frac{1}{2} \int_{\alpha}^{\beta} r(\$$theta)^2$$ \, d	heta. We $$will use this formula to find the areas inside each curve. Determine which curve is inside the other in the interval between the points of intersection. The overlapping area is the integral of the smaller radius squared over the appropriate interval. Set up the integral for the area of the region inside both curves accordingly. Finally, evaluate the integral(s) over the determined limits to find the area of the overlapping region. Remember to consider symmetry if applicable to simplify the calculation.

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

The region inside the rose r = 4 sin 2θ and inside the circle r = 2

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Concetti chiave

Polar Coordinates and Graphs

Area Calculation in Polar Coordinates

Finding Intersection Points of Polar Curves