Juan is designing a pool and a deck for his family. The pool has the shape of a cardioid whose equation is given by r = 4 - 4sinθ where r is in meters and θ is a number between 0 and 2π. The deck will be built in the region that lies inside the circle x2+ y2 = 4 and outside the cardioid. a. Convert the equation of the circle x2+ y2 = 4 in polar coordinates. b. Sketch the polar graphs and shade the region described in the problem. c. Find the polar point(s) where the cardioid and the circle intersect. d. Write an expression involving integrals that gives the area of the region where the deck will be built. Do not evaluate your expression.

Juan is designing a pool and a deck for his family. The pool has the shape of a cardioid whose equation is given by r = 4 - 4sinθ where r is in meters and θ is a number between 0 and 2π. The deck will be built in the region that lies inside the circle x2+ y2 = 4 and outside the cardioid. a. Convert the equation of the circle x2+ y2 = 4 in polar coordinates. b. Sketch the polar graphs and shade the region described in the problem. c. Find the polar point(s) where the cardioid and the circle intersect. d. Write an expression involving integrals that gives the area of the region where the deck will be built. Do not evaluate your expression.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 42E

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Juan is designing a pool and a deck for his family. The pool has the shape of a cardioid whose equation is given by r = 4 - 4sinθ where r is in meters and θ is a number between 0 and 2π. The deck will be built in the region that lies inside the circle x²+ y² = 4 and outside the cardioid.

a. Convert the equation of the circle x²+ y²= 4 in polar coordinates.

b. Sketch the polar graphs and shade the region described in the problem.

c. Find the polar point(s) where the cardioid and the circle intersect.

d. Write an expression involving integrals that gives the area of the region where the deck will be built. Do not evaluate your expression.