Triple Integral in Cylindrical Coordinates

Triple Integrals in Cylindrical coordinates Cylindrical coordinates of the point P: (𝑟,𝜃,𝑧) 𝑟 and 𝜃 are the polar coordinates of the projection of the point P onto the 𝑥𝑦-plane. 𝑧 is the signed vertical distance between P and the 𝑥𝑦-plane (same as in cartesian) From Cylindrical to Cartesian: From Cartesian to Cylindrical:

Triple Integrals in Cylindrical coordinates Example 1: Given find x, y and z. Conversely, given rectangular (−3,−3,7) find cylindrical: is a possible answer

Triple Integrals in Cylindrical Coordinates Basic graphs in cylindrical coordinates: 𝑟=𝑐 represents a cylinder ( 𝑥 2 + 𝑦 2 = 𝑐 2 in cartesian) 𝜃=𝑐 represents a vertical plane (if r ≥ 0, half a plane) 𝑧=𝑐 represents a horizontal plane 𝑧=𝑟 represents the cone 𝑧= 𝑥 2 + 𝑦 2

Triple Integrals in cylindrical coordinates Since 𝑥 2 + 𝑦 2 = 𝑟 2 , integrals involving 𝑥 2 + 𝑦 2 or 𝑥 2 + 𝑦 2 frequently are easier in cylindrical coordinates. The volume element is Theorem: (Change of coordinates) Let E be the region: Then the triple integral of f over E in cylindrical coordinates is

Triple Integrals in cylindrical coordinates – Example 2 Evaluate where E lies above z = 0, below z = y and inside the cylinder 𝑥 2 + 𝑦 2 =9. The plane 𝑧=𝑦 in cylindrical coordinates is 𝑧=𝑟 sin 𝜃 The domain D is the semicircle of radius 3: 0≤𝑟≤3, 0≤𝜃≤𝜋 The integrand function 𝑦𝑧 in cylindrical coordinates is (𝑟 sin 𝜃)𝑧

Triple Integrals in Cylindrical Coordinates – Example 3 Find the volume of the cone 𝑧= 𝑥 2 + 𝑦 2 for 𝑧≤4 using cylindrical coordinates. The cone and the plane 𝑧=4 intersect in a circle: This circle defines the boundaries for 𝑟 and 𝜃: 0≤𝑟≤4, 0≤𝜃≤2𝜋 In cylindrical coordinates the cone has equation 𝑧=𝑟, thus 𝑟≤𝑧≤4

Triple Integrals in Cylindrical Coordinates – Example 4 Sketch the solid whose volume is given by the integral and evaluate the integral. The solid is bounded below by 𝑧=0 (the 𝑥𝑦-plane) and above by the paraboloid 𝑧=9− 𝑟 2 =9− 𝑥 2 − 𝑦 2 The solid is bounded by the cylinder 𝑟=2 ( in cartesian: 𝑥 2 + 𝑦 2 =4) The solid is in the first octant (𝑥≥0 and 𝑦≥0)