Question

Finding Polar AreasFind the areas of the regions in Exercises 9–18.Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

Accepted Answer

First, identify the curves given in polar coordinates: the circle is described by $$r = 4 \sin 	heta$$ and the line by $$r = 3 \csc 	heta. $$Understand that $$\csc 	heta = \frac{1}{\sin 	heta}$$, so the line can be rewritten as $$r = \frac{3}{\sin 	heta}. $$Next, determine the region of interest: the area inside the circle $$r = 4 \sin 	heta$$ and below the line $$r = 3 \csc 	heta. $$This means we want the set of points where $$r is $$less than or equal to $$4 \sin 	heta$$ and also less than or equal to $$3 \csc 	heta$$, with $$r$$ values corresponding to points below the line. Find the points of intersection between the two curves by setting $$4 \sin 	heta = 3 \csc 	heta. $$Rewrite $$\csc 	heta as$$ $$\frac{1}{\sin 	heta} to $$get $$4 \sin 	heta = \frac{3}{\sin 	heta}. $$Multiply both sides by $$\sin 	heta to $$obtain $$4 \sin^2$ 	heta = 3. $$Solve for $$\sin 	heta to $$find the values of $$	heta$$ where the curves intersect. Determine the limits of integration for $$	heta$$ based on the intersection points and the region described (inside the circle and below the line). These limits will define the angular bounds for the area integral. Set up the integral for the area in polar coordinates using the formula $$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2$ \, d	heta. $$Since the region is bounded by two curves, express the area as the integral of the difference of the squares of the radii: $$A = \frac{1}{2} \int_{\alpha}^{\beta} \left( (4 \sin \$$theta)^2$$ - (3 \csc \$$theta)^2$$ ight) d	heta. $$This integral will give the area of the region inside the circle and below the line.

Finding Polar Areas

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

Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

Risposta video verificata per un problema simile:

Concetti chiave

Polar Coordinates and Graphing

Area Calculation in Polar Coordinates

Determining Intersection Points and Limits of Integration