Definition: In the c ylindrical c oordinate system, a point P in three-dimensional space is represented by the ordered triple (r, θ, z), where r and θ are polar coordinates of the projection of P onto the xy-plane and z is the directed distance from the xy-plane to P . Conversion From Cylindrical Coordinates to Rectangular Coordinates x = r cos θ y = r sin θ z=z Conversion From Rectangular Coordinates to Cylindrical Coordinates r 2 = x2 + y 2 tan θ = y x z=z Exercise 1. (a) Plot the point (2, 2π/3, 1) and find its rectangular coordinates. (b) Find cylindrical coordinates of the point (3, −3, −7). (Stew Sec 15.7 Ex 1) Class Exercise 1. Plot the point whose cylindrical coordinates are given. Then find the rectan√ (b) (1,1,1) (#2) gular coordinates of the point. (a) ( 2, 3π/4, 2) Class√Exercise 2. Change from rectangular coordinates to cylindrical coordinates. (a) (2 3, 2, −1) (b) (4, −3, 2) (#4) Exercise 2. Change the equation z 2 = x2 + y 2 to cylindrical coordinates, and sketch its graph. (Swok Sec 1.7 Ex 1) Exercise 3. Identify and sketch the following sets in cylindrical coordinates. (a) Q = { (r, θ, z): 1 ≤ r ≤ 3, z ≥ 0 } (b) S = { (r, θ, z): z = 1 − r, 0 ≤ r ≤ 1 } (Briggs Sec 13.5 Ex 1) Class Exercise 3. Describe in words the surface whose equation is given: r = 5. (#6) Class Exercise 4. Identify the surface whose equation is given: 2r2 + z 2 = 1. (#8) Class Exercise 5. Write the equations in cylindrical coordinates. (#10) (a) 3x + 2y + z = 6 (b) −x2 − y 2 + z 2 = 1 Class Exercise 6. Sketch the solid described by the given inequalities: 0 ≤ θ ≤ π/2, r ≤ z ≤ 2. (#12) Formula for Triple Integration in Cylindrical Coordinates: Suppose that E is a type I region whose projection D onto the xy-plane is conveniently described in polar coordinates: RRR R β R h (θ) R u (r cos θ,r sin θ) f (x, y, z) dV = α h12(θ) u12(r cos θ,r sin θ) f (r cos θ, r sin θ, z) r dz dr dθ E p Exercise 4. Find the volume of the solid D between the cone z = x2 + y 2 and the inverted paraboloid z = 12 − x2 − y 2 . (Briggs Sec 13.5 Ex 4) p Exercise 5. A solid Q is bounded by the cone z = x2 + y 2 and the plane z = 2. The density at P (x, y, z) is directly proportional to the square of the distance from the origin to P . Find its mass. (Swok Sec 17.7 Ex 4) Class Exercise RRR 7. Use cylindrical coordinates. (#18-24 even, 28) 2 (a) Evaluate z dV , where E is enclosed by the paraboloid z = x + y 2 and the plane z = 4. E RRR (b) Evaluate x dV , where E is enclosed by the planes z = 0 and z = x + y + 5 and by the E cylinders x2 + y 2 = 4 and x2 + y 2 = 9. (c) Find the volume of the solid that lies within both the cylinder x2 + y 2 = 1 and the sphere x2 + y2 + z2 = 4 (d) Find the volume of the solid that lies between the paraboloid z = x2 + y 2 and the sphere x2 + y2 + z2 = 2 (e) Find the mass of a ball B given by x2 + y 2 + z 2 ≤ a2 if the density at any point is proportional to its distance from the z-axis.