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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 53E

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Kindly answer letter B. Set up an integral for the perimeter of the shaded region. Show all solutions and explanations.

Let C₁ be the circle r = 3 cos 0 and C₂ the cardioid r = 1 + cos 0.
O
(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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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.
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 below
Find 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 loop of this polar curve: r=4costheta-2sectheta from -pi/3 to pi/3 using the formula A= 1/2 integral from -pi/3 to pi/3 (r)^2 dtheta.
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.
Consider a curve represented by x2 + y2 = 4x.(a) Find the polar equation of f.(b) Set up the integral to find the area inside the curve f and outside r = 2.

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