3. Let C1 be the circle r = 3 cos 0 and C2 the cardioid r = 1+ cos 0. %3D (a) Find all point(s) of intersection of the two given polar curves. (b) Set up an integral for the perimeter of the shaded region. (c) Set up an integral for the area of the shaded region.
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- Find the area of the region enclosed by one loop of the curve for the polar equation r = 2sin3θ.Use the integration capabilities of a graphing utility to approximate the area of the surface formed by revolving the polar equation over the given interval about the polar axis. r = 4 cos 2θ, [0, π/ 4]Given two polar curves and r=2cos2θ , r =1 as in Figure 1. Find the area of the shaded region by using the single integration in polar coordinates.
- Let C1 be the circle r = 3 cos θ and C2 the cardioid r = 1 + cos θ.(a) Find all point(s) of intersection of the two given polar curves. (b) Set up an integral for the perimeter of the shaded region. No need to evaluate.(c) Set up an integral for the area of the shaded region. No need to evaluate. Show necessary steps and process.2. Consider the polar curves r = 4 - 2cosθ and r = 2 + 2cosθ. In this problem, we want to find the area of A, B, and C pictured below. (c) B is the area inside both r = 2 + 2cosθ and r = 4 - 2cosθ. Find the area of B. (Hint: What happens at the angle where the two polar curves intersect? Your answer should involve a sum of two polar integrals.)II. Consider the circle C1 : r = 1 and the roses C2 : r = cos 2θ and C3 : r = 2 cos 2θ, each of which is symmetric with respect to the polar axis, the π/2-axis, and the origin, as shown on the image. 1. Find polar coordinates (r, θ) for the intersection A of C1 and C3, where r, θ > 0. 2. Set-up (do not evaluate) a sum of three definite integrals that give the perimeter of the yellow-shaded region inside both C1 and C3 but outside C2. 3. Find the area of the unshaded region inside C3 but outside C1.
- Sketch the regions defined by the polar coordinate inequalities 0 ≤ r ≤ 6 cos θFind the area enclosed by one loop of this polar curve: r=3sqrt(cos2theta) from 0 to 2pi using the formula A=1/2 integral from 0 to 2pi (r)^2 for parametric curve.Find the area enclosed by the curve r = 3sin 2θ defined in polar coordinates.
- Consider polar curves C1 : r = −3 sin(2θ) and C2 : r = 3 sin θ.Set up the definite integral for the perimeter and area of the region outside C1 but inside C2. See graph belowFind the area inside the cardioid with a polar curve of r=4+4cos(θ).Find the area of the region which is inside the polar curve r=3cos(theta) and outside the curve r=2-1cos(theta)